# Interface magnetic moments enhancement of FePt-L10/MgO(001): an $ab$
$initio$ study

R. Cuadrado  Department of Physics, University of York, York YO10 5DD, United
Kingdom  R. W. Chantrell  Department of Physics, University of York, York YO10
5DD, United Kingdom

###### Abstract

The interface between FePt–L10 and MgO(001) alloys has been studied using
density functional calculations. Because the stacking of the face-centered
tetragonal L10 phase is formed by alternating Fe and Pt planes, both the Fe-
and Pt-terminated contact layers were studied. Furthermore, due to the large
mismatch between the in-plane lattice constants of both systems, we have
chosen some common $a$ values for both alloys in order to explore in detail
the adsorption energy, the electronic structure and the interface magnetism.
The adsorption energy has been calculated by subtracting the energy of clean
FePt and MgO alloys from the total energy. The preferred adsorbed geometric
sites for Fe/Pt atoms are when they lie on $top$ of the O species, having a
smaller adsorption energy for the remaining positions. We found that expanding
the MgO lattice enhances the magnetic moment of the Fe species but the Pt
moments remain almost constant.

##  I Introduction

The face-centered tetragonal (fct) L10 phase of the 3d–5d binary based alloys
such as FePt have recently been the subject of much attention because of their
potential applications for the fabrication of ultrahigh density data recording
media weller_para . These alloys present high values of the magnetocrystalline
anisotropy (MAE) constants (7$\times$107 ergs/cm3) Pirama ; ivanov  along the
$c$–axis, a preferred orientation direction (easy axis). These high values of
the anisotropy are necessary to overcome the superparamagnetic limit Yan  in
order to avoid the loss of recorded information. There are several methods to
achieve the desired ferromagnetic structures such as the alternating monatomic
layer deposition of Fe and Pt Shima  or alternatively the room temperature
deposition of disordered fcc FePt on an underlayer followed by annealing at
around 600 ∘C to induce a phase transformation from fcc to fct L10 stacking.
Extensive studies such as the effect of the alloying composition and growing
FePt films on various underlayers, for instance MgO, Chen  PtMn, Chiang  or
Si, Wu  among others, have been carried out to optimize the microstructure and
the magnetic properties as well as decreasing the processing temperature Xu .
For practical purposes there are still some challenges, for example the
ordered FePt grains with perpendicular (001) crystallographic orientation have
to be magnetically decoupled from each other for which some materials are
added in the fabrication process (see Ref. Peng  and references therein). The
use of MgO as an underlayer has some practical drawbacks such as its elevated
costs, but the MgO single crystalline substrate promotes the out-of-plane
anisotropy in contrast to other substrates which tend to promote in-plane
anisotropy. Perumal  Additionally, because the MgO lattice parameter is larger
than that of FePt, it can easily promote the $c_{FePt}$ to remain
perpendicular during the growing process, thus reducing the in-plane variants.

From a theoretical point of view the transition metal (TM) binary compounds
have been extensively studied in the past decade in their bulk phases, slabs,
Chepulskii2012  gas phases, Cuadrado ; antoniak ; rollman  and interfaces Zhu1
, however only a few theoretical studies based on FePt–L10/MgO(001) interfaces
have been carried out Zhu1 ; Zhu2 . In this work we scan different
configurations for this interface in a systematic investigation of the
potential adsorption geometries, namely, the Fe-/Pt-termination, different
atomic adsorption sites and the influence within the electronic structure of
changes in the MgO(001) lattice constant, $a_{MgO}$.

The paper is structured as follows: The employed theoretical tools are
explained briefly in section II. In section III.1 we will summarize the final
geometric structures as well as the related adsorption energies. The
electronic study is presented in the section III.2 and the magnetic behaviour
in III.3. The conclusions and future work are in section IV.

##  II Theoretical Methods

Figure 1: (Color online) (A) Schematic top view of the four initial adsorption
sites for FePt–L10 onto MgO(001) alloy. Only the atoms belonging to the
interfaces are shown, i.e., big green and small red spheres for the MgO
contact layer and blue spheres on whether Fe- or Pt-terminations is shown. (B)
FePt–L10 and MgO unit cells, left and right, respectively. The in-plane
lattice constant, $a$, as well as some representative orientations and values
have been also depicted in the figure.

We have undertaken geometrical, electronic and magnetic structure calculations
of the FePt–L10/MgO(001) interface by means of DFT using the SIESTA siesta
code. To describe the core electrons we have used fully separable Kleinmann-
Bylander kb  and norm-conserving pseudopotentials (PP) of the Troulliers-
Martins tm  type. Our DFT based calculations have been performed within the
generalized gradient approximation (GGA) for the exchange correlation (XC)
potential following the Perdew, Burke, and Ernzerhof (PBE) version pbe . To
address the description of magnetic systems, pseudocore (pc) corrections were
used to include in the XC terms not only the valence charge density but also
the core charge cc . In order to ease the convergence of three center
integrals with the size of the real space grid, $\rho^{c}(r)$ is replaced by a
pseudo-core charge density, $\rho^{pc}(r)$, which equals the real core charge
density beyond a given radius, $r_{pc}$, while close to the nuclei it becomes
a smooth function. The radius $r_{pc}$ should be chosen small enough to ensure
that the overlap region between the valence and the core charges is fully
taken into account. Based on previous studies of the binary alloys LS-paper ,
we have chosen for this radius the values of rpc(Fe) = 0.6 Bohrs and rpc(Pt) =
1.0 Bohrs, ensuring that the overlap region between the valence and the core
charge is fully taken into account. As the basis set, we have employed double-
zeta polarized (DZP) strictly localized numerical atomic orbitals (AO). The
confinement energy, $E_{c}$, defined as the energy cost to confine the wave
function within a given radius, was set to 100 meV. The so–called electronic
temperature –kT in the Fermi-Dirac distribution– was set to 50 meV. In all the
cases we ensured convergence of the Brillouin Zone (BZ) integration by
considering a k–supercell of (16x16), i.e., 256 k-points. Real space three-
center integrals are computed over a three-dimensional grid with a resolution
of 900 Ry, a mesh fine enough to ensure convergence of the magnetic
properties.

The interface system is described by a two-dimensional periodic slab
comprising eight MgO(001) plus eight FePt–L10 oriented layers. The L10
structure stacking is a fct phase in which there are alternate planes of Fe
and Pt along the (001) direction. Because of this, there arises the
possibility of having two kinds of interfaces between FePt–L10 and MgO(001):
Fe-terminated (Fe/MgO) and Pt-terminated (Pt/MgO). Furthermore we can see in
the Fig. 1A schematically the top views of four possible initial
configurations, first and second rows. The small red and big green spheres
represent the MgO alloy and the blue represent the first contact plane of the
FePt alloy, either Fe or Pt. On the other hand, in the Fig. 1B the unit cells
are shown following the same atomic nomenclature as in A but now explicitly
the Pt atom presents a bigger size than the Fe ones. Specifically, depending
whether the Fe/Pt atoms lie on $top$, $hollow$ or at $bridge$ positions, we
have named the configurations as follows: on $top$ of O (Fe/Pt@$top$-O), on
$top$ of Mg (Fe/Pt@$top$-Mg), at $hollow$ (Fe/Pt@$hollow$) and in $bridge$
positions (Fe/Pt@$bridge$). Metastable adsorption structures were obtained
after relaxing the different proposed models to local minima until forces on
atoms were smaller than 0.03 eV/Å. During the minimization process, just two
layers of each material were allowed to relax leaving the rest of the atoms in
the slab fixed to their bulk positions.

The presence of two kinds of atoms in the FePt-L10 phases generates a vertical
distortion so that its structure is defined by two quantities, the in-plane
lattice parameter, $a$, and the out-of-plane constant, $c$, whose bulk
experimental values are $a_{FePt}$=3.86 Å and $c/a$=0.98. The magnesium oxide
structure can be described as two inter–penetrating fcc lattices displaced by
$a/2(111)$ along the body diagonal of the conventional cube and the bulk
experimental value for its lattice parameter is $a_{MgO}$=4.22Å. We observe
that the in-plane mismatch between both alloys is $\approx$8.5$\%$. Because of
this and in order to scan more geometrical possibilities we have used the four
common in-plane lattice values: 4.00Å, 4.05Å, 4.10Å and 4.30Å. The last value
corresponds to the optimized lattice constant for the bulk MgO under GGA and
it has been taken into account to address how the FePt geometry and magnetic
properties change not only with intermediate common $a$ values but also with
an $a_{MgO}$ optimized value. According to the changing $a$, the distance
between planes will vary too, so it was necessary to optimize the $c$
parameter for each lattice value. The results of both bulk systems and their
corresponding out-of-plane distortions are for MgO: c/a = 1.14, 1.12, 1.10,
1.00 and for FePt: c/a = 0.92, 0.90, 0.88, 0.75 for a = 4.00Å, 4.05Å, 4.10Å
and 4.30Å, respectively.

##  III Results

###  III.1 DFT structural relaxations

|  |  | Fe-terminated |  |  | Pt-terminated |
---|---|---|---|---|---|---|---
Site | a (Å) |  | Eads | $z_{I}(Fe)$ | MMM | MMNM |  |  | Eads | $z_{I}(Pt)$ | MMM | MMNM |
@$top$-O | 4.00 |  | 0.89 | 2.23 | 3.22 | 0.24 |  |  | 0.57 | 2.55 | 3.26 | 0.25 |
| 4.05 |  | 0.93 | 2.21 | 3.26 | 0.25 |  |  | 0.60 | 2.57 | 3.30 | 0.26 |
| 4.10 |  | 0.97 | 2.19 | 3.29 | 0.25 |  |  | 0.62 | 2.56 | 3.34 | 0.26 |
| 4.30 |  | 1.14 | 2.16 | 3.34 | 0.21 |  |  | 0.74 | 2.47 | 3.39 | 0.20 |
@$top$-Mg | 4.00 |  | 0.19 | 3.30 | 3.26 | 0.25 |  |  | 0.22 | 3.16 | 3.27 | 0.25 |
| 4.05 |  | 0.20 | 3.47 | 3.30 | 0.25 |  |  | 0.22 | 3.13 | 3.30 | 0.25 |
| 4.10 |  | 0.20 | 3.51 | 3.33 | 0.25 |  |  | 0.23 | 3.14 | 3.34 | 0.26 |
| 4.30 |  | 0.22 | 3.54 | 3.37 | 0.21 |  |  | 0.29 | 3.07 | 3.40 | 0.21 |
@$hollow$ | 4.30 |  | 0.57 | 2.47 | 3.36 | 0.21 |  |  | 0.45 | 2.68 | 3.40 | 0.21 |
@$bridge$ | 4.30 |  | 1.13 | 2.17 | 3.34 | 0.21 |  |  | 0.70 | 2.51 | 3.39 | 0.20 |
Table 1: Adsorption energies, Eads, $z$ heigths between the MgO contact layer
and the first FePt-L10 plane, $z_{I}(Fe/Pt)$, average magnetic moments (MM)
per atom of the Fe and Pt species of each configuration for Fe-/Pt-
terminations, columns 3 to 6 and 7 to 9, respectively. The first two columns
represent the four adsorption sites and the common lattice in-plane $a$
values, respectively. Energies are in eV, heights in Å and MM in $\mu_{B}$/at.

In the table 1 we present the results of the interface adsorption energies
Eads, the perpendicular bond distance $z_{I}$ computed as the $z$ difference
of the MgO plane and the Fe/Pt one: $z_{I}$(Ai)=$z_{A_{i}}$–$z_{B_{i}}$
[Ai=Fe,Pt; Bi=Mg,O] and the total magnetic moment (MM) per magnetic/non-
magnetic species for all the configurations. Two different contact layers,
namely, Fe- and Pt-termination have been taken into account with a common in-
plane lattice constant ranging from 4.00 to 4.30Å. The adsorption energies
were evaluated after subtracting from the total energy of the FePt+Mg(001)
bilayer the energy of the two clean metallic slabs.

A general tendency of all @$top$ relaxed structures is that as the in-plane
lattice constant expands from 4.00Å to 4.30Å the Eads increases significantly.
It is noticeable that for the Fe-terminated configuration on top of O (first
four values in the third column of Table 1), the average adsorption energies
are larger by 0.35, 0.87 and 0.74 eV compared to those of Pt@$top$-O,
Fe@$top$-Mg and Pt@$top$-Mg sites, respectively. Consequently, the bond
between the FePt and MgO will be stronger for the Fe@$top$-O, with decreasing
stability for the other cases, with Fe@$top$-Mg having the lowest stability.
In order to scan more possible accommodation sites for Fe/Pt atoms we also
studied Fe/Pt@$hollow$ and Fe/Pt@$bridge$ adsorption configurations (see Fig.
1A) considering only the GGA optimized value of $a_{MgO}$. The Eads values in
these cases are between of those of Fe/Pt@$top$-O/-Mg sites, providing a weak
chemical bonding for FePt. Regarding the Fe/Pt@$bridge$ case and after
relaxing the interfaces, we found that the strong Fe/Pt-O chemical interaction
means that the Fe/Pt@$bridge$ atoms move closer to the @$top$-O positions. In
fact, the equilibrium position for the Fe/Pt atoms is only $\sim$0.2 Å distant
from their @$top$-O positions.

We also notice, by inspection of the table, that Eads values are quite similar
in both cases. Related to the Eads, the bond distance, $z_{I}$(Ai), will
depend on the strength of the bond: the higher the adsorption energies the
smaller $z$ distances and stronger bonding will arise. This implies that the
Fe/Pt planes will be closer to the MgO contact layer resulting in a complex
rearrangement of the charge, changing the final values of the magnetic moments
and hence the magnetic behaviour. Except for the case of Fe@$top$-Mg, where
Eads remains approximately constant, the interlayer bond distance decrease as
$a$ increases. We will discuss in detail in section III.3 the behaviour of the
magnetic moments. As a global tendency we can say that the MMM values are
augmented as $a$ is increased, by an average amount of $\sim$0.13$\mu_{B}$ and
remain almost constant for MMNM with the special characteristic reduction of
these values for $a$=4.30Å.

###  III.2 Density of states and hybridisation study

We display in Fig. 2 the evolution of the spin-resolved density of states
(DOS), projected on the Fe/Pt interface atoms, for Fe- and Pt- terminated FePt
layers. For both cases results are presented for the @$top$-O and @$top$-Mg
adsorption sites. The next FePt–L10 layers after the contact layer (Pt atoms
for Fe-termination or Fe for Pt-termination) are not shown here because
changes in the electronic states are not significant beyond the interface
layer. The three different colours in each graph depict the Fe/Pt DOS of the
bilayer (filled turquoise curve), the Fe/Pt for the clean surface (thick solid
pink line) and the atomic DOS projection of the atoms in their bulk phase
(thin solid black line). Inside each graph the MM values appear on the bottom
left corner just for the bilayer configurations together with a positive
(negative) value within parenthesis that represents the increase (decrease) of
the local MM compared to the Fe/Pt atomic bulk values.

Figure 2: (Color online) Density of states (DOS) projected on Fe/Pt atoms of
the FePt–L10 interfaces (filled turquoise curve), clean surfaces (thin solid
pink line) and the Fe/Pt atoms in their bulk phases (thin solid black line).
The Fe-/Pt-terminated configurations are presented in the first and second and
in the third and fourth columns, respectively. For each interface termination
@$top$-O/-Mg adsorption sites are also shown. The numbers present the MM in
$\mu_{B}$/at and in parenthesis their difference with respect to the bulk
atomic values.

A first inspection of Fig. 2 shows that the DOS at the interface layer is
significantly affected for contact with both MgO and vacuum. Further
inspection of the DOS for the four rows (O and Mg adsorption sites) regarding
both the Fe-/Pt-terminations, shows that, when the Fe/Pt atoms lie @$top$-Mg
rather than @$top$-O, the interface Fe/Pt PDOS profile is almost the same as
those of clean surfaces either spin-up or spin-down. Consequently the surface
states of the FePt alloy will not be altered significantly. As a result, when
Fe/Pt are @$top$-Mg the FePt termination behave similarly to a vacuum
termination irrespective of the contact with MgO. This confirms the
significant difference of $\sim$0.74 eV between the adsorption energies
regarding the Mg or O sites (see table 1). Then, as we pointed out in the
previous section, the bonding between Fe/Pt and Mg atoms will be weaker than
with the O atoms.

The abrupt termination of the FePt–L10 alloy in the clean surfaces and in the
interface of the bilayer induces within the $d$ electrons a rearrangement
which is distinct from that in the bulk phase Although the Fe/Pt coordination
will be the same for the whole system, the environment will be modified
compared to that of the bulk, given that now the Mg and O atoms are in the
former positions of the Fe/Pt species. This is noticeable if we inspect the
DOS of the Fe-termination (first two columns). In the bulk case the up-states
are located in an energy range of 5 eV, i. e., from -1 to -6 eV for the
@$top$-O/-Mg configurations. Despite the fact that the DOS of the interfaces
and clean surfaces are quite similar to those of the bulk, their states are
located closer to a pronounced peak at 3 eV weakly present in the bulk,
leading to a narrowing of both the bilayer and clean surface d-bands.

The fact that the spin-up and spin-down black curves have a small shift to
lower energy values, compared to the pink and turquoise curves, in the last
two columns of Fig. 2 (Pt-termination) can be explained by inspection of the
Mulliken population of the bulk FePt. This shows that the bulk atomic species
are more charged than the other two systems by an average of $\sim$0.2 e/at.

Figure 3: (Color online) Fe and Pt atomic charge difference with respect to
their bulk phase counterparts, $\Delta
Q_{Fe/Pt}=q_{Fe/Pt}^{bilayer}-q^{bulk}_{Fe/Pt}$, as a function of the in-plane
lattice values for the Fe-/Pt-terminated configurations, left and right,
respectively. The solid coloured lines depict the evolution of $\Delta
Q_{Fe/Pt}$ interface layer @top-O/-Mg adsorption sites and the dashed ones
those of the nearest inward layer: Pt for Fe-terminations and Fe for Pt-
termination.

The analysis of the data shows that the charge transfer between the atoms
(orbitals) belonging to an interface is of paramount importance. It gives us a
way to understand the hybridisation between atoms and how this influences the
bond strength and the magnetic behaviour. In figure 3 it shown that the
difference in charge transferred between the magnetic and non–magnetic species
is larger compared to the bulk phases. In figure 3A, situating the Fe atoms in
the @$top$-O position (blue solid line) increases the amount of charge given
to the Pt layer (dashed blue line) and to the first MgO plane by an average of
0.18 e/at. The same situation occurs when the Fe is @$top$-Mg (solid green
line) though the effect is smaller. The dispersion ranges from 0.07 e/at up to
almost 0.2 e/at for 4.30Å. When the Pt contact layer is @$top$-O (black solid
line) it behaves in a similar way to bulk phase so that only for the 4.30Å
lattice spacing is the amount of excess charge transferred to the Pt is
significant at 0.08 e/at. Similar behavior is shown when comparing the Fe
contact configuration in A. The Fe atoms in this case (black dashed line) are
responsible for the charge transfer, having almost a constant contribution
along all the $a$ values. Finally, the charge transferred to the Pt atoms
@$top$-Mg (red solid line) shows very little change with respect to the bulk
phase, increasing only slightly for the large $a$ values. The increase is
clear however for the Fe atoms: again they lose more charge than in the bulk.
In summary, the presence of the MgO changes significantly the FePt-L10
behavior depending on whether the contact layer is Fe or Pt and also where the
Fe/Pt atoms are located after relaxation. This implies different kinds of
hybridisation between 3$d$ and 5$d$ orbitals of these Fe and Pt atoms
respectively.

In order to link the MM behaviour with the DOS curves we also observe that in
Table 1 the MM values of the Fe atoms increase by 0.13$\mu_{B}/at$ with the
in-plane lattice spacing for all the configurations. This small enhancement of
the MMM can be observed in the DOS curves noting that the down-state peaks
below the Fermi level at $\sim$0.2 eV for $a=$4.00Å move to lie just at the
Fermi level for $a=$4.30Å resulting in a deficit of down-states compared to
the up-states which are almost constant with $a$. Finally, the DOS projected
onto the interface plane of the MgO alloy (not shown here) does not change
significantly, there is only some charge transfer among the interface
resulting in a very small value of the local MM values (see Sec. III.3).

###  III.3 Interface magnetism: Magnetic Moments

Figure 4: (Color online) Resolved magnetic moments of Pt and Fe atoms, first
and second row, respectively, for both Fe- (left) and Pt-terminations (right)
as a function of the in-plane lattice parameter $a$ (in Å). For a common type
of symbol the empty symbols represent the MM of the non magnetic species and
the filled symbols, those of the Fe atoms. Each kind of color depicts @$top$-O
adsorption site (blue squares), @$top$-Mg (green squares), the bulk (red
circles) and the clean surface (orange triangles). The crosses and asterisks
show the $bridge$ and $hollow$ adsorption sites for 4.30Å, respectively.

We provide in Fig. 4 the magnetic moment (MM) values per atom as a function of
the in-plane lattice parameter $a$ for all the configurations studied in this
work. The global trend of the MM values of the Fe atoms for both Fe-/Pt-
terminations (bottom row) is an enhancement of around 0.13$\mu_{B}$/at as the
lattice increases from 4.00Å to 4.30Å. Although this behaviour is shared by
all the adsorption positions, i.e., @(top-O, top-Mg,hollow,bridge) there is
just an increase of 0.06$\mu_{B}$/at in the bulk case (filled red circles)
when $a$=4.30Å compared to the smaller lattice considered. The values of the
MM of the Pt atoms for both Fe-/Pt-terminations (first row) are enhanced as
are those of the Fe atoms when $a$ increases up to $a$=4.15Å. However the MM
decreases (increases) for the Fe-terminated alloys (Pt-terminated) for larger
$a$. Inspecting the MM values vertically (common lattice) either @top-O or
@top-Mg, blue and green filled squares, respectively, we observe that the
dispersion between these two adsorption sites is about 0.15$\mu_{B}$/at. It is
evident after inspecting table 1 that the lower distance between the Fe/Pt:MgO
layers gives the higher adsorption energy. This behaviour plays an important
role in the net MM/at values since the charge rearrangement between down and
up states of the system promotes the reduction or the increase of the MM/at
depending if the adsorption site is @top-O or @top-Mg, respectively.
Completing the magnetic study we have added dots to represent the MM values
for the Fe/Pt@($hollow$,$bridge$) sites, black asterisks and red crosses,
respectively. Their MM values do not have significant changes regarding others
with same $a$ value, they are almost the same as those obtained for the
Fe/Pt@$top$-O configurations.

|  |  | Fe-terminated |  |  | Pt-terminated |
---|---|---|---|---|---|---|---
Site | a(Å) |  | MMMg | MMO |  |  | MMMg | MMO |
@$top$-O | 4.00 |  | -0.11 | 0.03 |  |  | -0.03 | 0.03 |
| 4.05 |  | -0.12 | 0.03 |  |  | -0.03 | 0.03 |
| 4.10 |  | -0.12 | 0.03 |  |  | -0.04 | 0.03 |
| 4.30 |  | -0.10 | 0.04 |  |  | -0.03 | 0.04 |
@$top$-Mg | 4.00 |  | -0.00 | 0.02 |  |  | -0.03 | 0.02 |
| 4.05 |  | -0.02 | 0.01 |  |  | -0.03 | 0.02 |
| 4.10 |  | -0.02 | 0.01 |  |  | -0.03 | 0.02 |
| 4.30 |  | -0.01 | 0.01 |  |  | -0.04 | 0.03 |
@$hollow$ | 4.30 |  | -0.05 | 0.04 |  |  | -0.02 | 0.04 |
@$bridge$ | 4.30 |  | -0.10 | 0.04 |  |  | -0.03 | 0.03 |
Table 2: Local magnetic moments (MM) values of the Mg (MMMg) and O (MMO) atoms
of the MgO contact plane in $\mu_{B}$. The nomenclature as well as the
different adsorption sites and terminations are the same as used throughout
this work.

Finally we show in Table 2 the site resolved MM values of Mg and O atoms. Note
that the Mg species favors the down states while the O sites exhibit the
opposite behavior. The most significant difference between the distinct
configurations occurs @$top$-O adsorption sites since the MMMg are almost
constant, at a value of -0.12 $\mu_{B}$ in comparison with the rest. The O
species have an average constant value of $\sim$0.03 $\mu_{B}$.

##  IV Conclusions and future work

We have carried out a first principles study of the FePt–L10/MgO(001)
interface regarding different in-plane lattice constant values and the two
possible FePt contact planes due to the L10 stacking. In addition we scanned
the magnetic/electronic properties for different adsorption sites for a common
Fe/Pt plane, namely, @($top$-O,$top$-Mg,$hollow$,$bridge$). The adsorption
energies provide a way to elucidate the preferential adsorption site and
lattice constant value. The higher values were obtained for Fe@$top$-O
followed by those of the Pt@$top$-O and also when the larger in-plane lattice
constant has been taken into account. The stronger chemical bonding then
occurs for these configurations having a significant overlap between the
$d$-bands and the MgO orbitals. Additionally, we found that as the Eads
decreases(increases) the MMM values augment(diminish) however the induced MM
within the Pt species is not significantly affected by this variation. The use
of GGA as the XC could overestimate the bonding between these materials giving
the possibility that from an experimental point of view, the bond distance
would be shorter and hence the adsorption energy higher.

A useful compliment to the discussion of atomic hybridisation at the end of
section III.2 would be a study of the up/down population of the $d$-orbitals.
It is however beyond the scope of this work to include such details and
instead we have only explored qualitatively the atomic spin-resolved
population to understand how the spin charge influences the magnetic behavior.
The spin electronic analysis confirms the results of the DOS calculations
(Fig. 2). It shows that with increasing lattice constant, $a$, the amount of
up (down) charge increases (decreases) leading to an enhancement of the MM
values.

In addition, figure 4 shows that there is a constant difference of
0.25$\mu_{B}$/at in the MM values for different adsorption positions (see
separate blue and green symbols in the Fe-terminated configuration). The
lowest adsorption energy value is shown for the Fe@$top$-O and hence this is
the most stable configuration. This means that the hybridisation between the
MgO and the Fe contact layer is stronger, reducing the MM values and promoting
a bigger charge transfer among atoms. For the Pt-terminated configurations, as
was pointed out in section III.1, the adsorption energy is smaller and the
orbital hybridisation is smoother, therefore the MM values are close to each
other for all values of $a$.

It has recently been shown fefept  that the site-resolved Magnetocrystalline
Anisotropy Energy (MAE) of FePt is associated with the Fe sites. This is due
to the strong 3d Fe – 5d Pt hybridisation through which the spin-orbit
interaction on the Pt atoms is transferred to the electronic states at the Fe
sites. Consequently, the complex charge transfer processes at the FePt/MgO
interface predicted here might be expected to be reflected in changes in the
FePt MAE at the MgO interface. However, the charge transfer effects seem
rather localized to the FePt/MgO interface, so the effect might be significant
only for ultrathin films. However, this is an interesting possible effect,
which is beyond the scope of the current work but certainly worthy of further
investigation.

##  V Acknowledgments

The authors are grateful to Dr T.J Klemmer for helpful discussions. Financial
support of the EU Seventh Framework Programme under grant agreement No.
281043, FEMTOSPIN, and Seagate Technology is gratefully acknowledged.

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# BME Submission for SIGMORPHON 2021 Shared Task 0. A Three Step Training
Approach with Data Augmentation for Morphological Inflection

Gábor Szolnok∗
Budapest University of
Technology and Economics
<EMAIL_ADDRESS>
Botond Barta∗
Budapest University of
Technology and Economics
<EMAIL_ADDRESS>
Dorina Lakatos∗
Budapest University of
Technology and Economics
<EMAIL_ADDRESS>
Judit Ács
Budapest University of
Technology and Economics
<EMAIL_ADDRESS>

###### Abstract

We present the BME submission for the SIGMORPHON 2021 Task 0 Part 1,
Generalization Across Typologically Diverse Languages shared task. We use an
LSTM encoder-decoder model with three step training that is first trained on
all languages, then fine-tuned on each language families and finally fine-
tuned on individual languages. We use a different type of data augmentation
technique in the first two steps. Our system outperformed the only other
submission. Although it remains worse than the Transformer baseline released
by the organizers, our model is simpler and our data augmentation techniques
are easily applicable to new languages. We perform ablation studies and show
that the augmentation techniques and the three training steps often help but
sometimes have a negative effect. Our code is publicly
available111https://github.com/szogabaha/SIGMORPHON2021-task0
∗ The first three authors contributed equally..

##  1 Introduction

Morphological inflection is the task of generating inflected word forms given
a lemma and a set of morphosyntactic tags. Inflection plays a key role in
natural language generation, particularly in languages with rich morphology.
The SIGMORPHON Shared Tasks are yearly competitions for inflection
tasksCotterell et al. (2016, 2017, 2018); McCarthy et al. (2019); Nicolai et
al. (2020).

This paper describes the BME team’s submission for Part 1 of the 2021
SIGMORPHON–UniMorph Shared Task on Generalization in Morphological Inflection
Generation. There were only two submissions to this subtask and our team
outperformed the other submission by a large margin. The task was about type-
level morphological inflection in 38 typologically diverse languages from 12
language families.

Our model builds on the work of Faruqui et al. (2015). We use a sequence-to-
sequence (seq2seq) model with a bidirectional LSTM Hochreiter and Schmidhuber
(1997) encoder and a unidirectional LSTM decoder with attention. We perform a
small hyperparameter search and we find that the most important parameters are
the choice of the embedding size and the hidden size.

We use two data augmentation techniques; a simple copy mechanism, a stem
modification method. We add these methods at the first two steps of our three-
step training regime. We describe a third data augmentation technique, a
template induction method, that did not improve the overall results in the end
so we did not use it in our submission. We first train a single model on all
languages and then fine-tune the model on each language family and then on
each language.

Our main contributions are:

  * •

We present the highest performing submission for the SIGMORPHON 2021 Task0
Part 1 shared task.

  * •

We try three data augmentation techniques.

  * •

We use a three-step training regime mixed with a two data augmentation
techniques applied at the first two steps.

  * •

We perform ablation studies for the augmentation techniques and the training
steps. We highlight the negative results as well.

##  2 Related Work

Seq2seq neural networks were first popularized in machine translation
Sutskever et al. (2014) and since the addition of the attention mechanism
Bahdanau et al. (2014); Luong et al. (2015) and the Transformer Vaswani et al.
(2017), it has been the dominant approach in the field. The first major
success of seq2seq models in morphological inflection was the submission by
Kann and Schütze (2016) to the 2016 edition of the SIGMORPHON shared task.
This was followed by an extensive study by Faruqui et al. (2015) on LSTM-based
encoder-decoder models for morphological inflection.

We used the augmentation techniques introduced by Neuvel and Fulop (2002).
Inspired by Bergmanis et al. (2017) we attempted to extract different
morphological properties of the languages and used them to generate data.
Anastasopoulos and Neubig (2019) used a two step training method that first
trains on the language family and then on the individual languages. We use a
similar procedure but we augment the data with a different technique in each
step.

##  3 Data

The shared task covered 38 languages from 12 language families. 35 of these
languages were available from the beginning while 3 surprise languages,
Turkish, Vibe and Võro, were released one week before the submission deadline.
Each language had a train and a development split of varying size. Each sample
consists of a lemma, an inflected form and a list of morphosyntactic tags in
the following format:
vaguear vaguearás V;IND;SG;2;FUT
emunger emunjamos V;IMP;PL;1;POS
desenchufar desenchufo V;IND;SG;1;PRS
delirar deliraren V;SBJV;PL;3;FUT

The amount of data varies widely among the languages: while the language Veps
has more than 100000 examples, Ludic, the most underresourced language, has
only 128 train samples. Table 1 lists the 12 language families and the number
of languages from that family. We consider some languages low resource
languages if they have fewer than 1300 samples. 8 language families had at
least one low resource language and 3 families were represented only by low
resource languages. One goal of our data augmentation techniques is to offset
this imbalance (see Section 5).

Family | Langs | Low | Samples
---|---|---|---
Afro-Asiatic | 12 | 1 | 196550
Arnhem | 1 | 1 | 214
Aymaran | 1 | 0 | 100000
Arawakan | 2 | 0 | 16472
Iroquoian | 1 | 0 | 3801
Turkic | 3 | 0 | 300371
Chukotko-Kamchatkan | 2 | 2 | 1378
Tungusic | 2 | 1 | 5500
Austronesian | 2 | 1 | 11395
Trans-New-Guinean | 1 | 1 | 918
Indo-European | 12 | 2 | 685567
Uralic | 5 | 2 | 279720

Table 1: List of language families and the number of languages from each
family. The third column is the number of low resource (<1300 samples)
languages in a particular family. The forth column is the overall sample count
in each family.

##  4 Model architecture

###  4.1 LSTM based seq2seq model

Our model is largely based on the encoder-decoder model of Faruqui et al.
(2015). We use a bidirectional LSTM Hochreiter and Schmidhuber (1997) as our
encoder and a unidirectional LSTM with attention as our decoder.

Recall that the input for the inflection task is a pair consisting of a lemma
and a list of morphosyntactic tags. We represent these pairs as a single
sequence as the LSTM’s input. For the input lemma-tags pair izar, (V, COND,
PL, 2), we serialize it as

<SOS> i z a r <SEP> V COND PL 2 <EOS>

Similarly, we convert the target form into a sequence of characters:

<SOS> i z a r <EOS>

The output of our model looks like this when the inflected word is izaríais:

<SOS> i z a r í a i s <EOS>

The input sequence is first projected to an embedding space, which then
provides the input for the encoder LSTM. The decoder is a standard
unidirectional LSTM with attention. We decode the output in a greedy fashion
and do not use beam search. We project the final output to the output
vocabulary’s dimension and use the softmax function to generate a probability
distribution. The input and the output embeddings use shared weights and they
are trained from scratch along with the rest of the model.

Family |  Lang code | Result | excluded feature | basemodels
---|---|---|---|---
Copy | Stem-mod | Step 1 | Step 2 | Step 3 | IIT+DA | OL
Turkic | tur | 99.90 | 99.90 | 99.94 | 99.92 | 97.38 | 99.90 | 99.35 | 97.10
Uralic | vep | 99.72 | 54.10 | 99.55 | 99.80 | 99.05 | 99.67 | 99.70 | 91.13
lud | 59.46 | 56.76 | 70.27 | 56.76 | 67.57 | 62.16 | 45.95 | 0.00
olo | 99.72 | 91.15 | 99.84 | 99.78 | 98.26 | 99.72 | 99.66 | 99.48
Indo - European | rus | 98.07 | 94.84 | 98.00 | 97.86 | 95.56 | 97.34 | 97.58 | 70.72
kmr | 98.21 | 86.02 | 98.74 | 98.41 | 97.50 | 98.21 | 98.01 | 5.14
deu | 97.98 | 91.19 | 98.23 | 97.91 | 89.91 | 97.98 | 97.46 | 91.86
Table 2: The different results we achieved on the test dataset with different
models, with different augmentation techniques excluded and with different
training steps excluded. For comparison the table show the result of our
submission (_result_), the given basemodel _IIT+DA_ (Input Invariant
Transformer + Data Augment) and the models that were just trained on only one
language (_OL_).

###  4.2 Hyperparameter selection

We selected 16 languages from diverse families for hyperparameter tuning. Most
of them were fusional or low resource because early experiments showed that
these are the harder ones to learn for the model. We downsampled the larger
languages and merged the train sets. We trained 100 models with random
parameters sampled uniformly from the parameter ranges listed in Table 3.

Parameter | Type | Min value | Max value
---|---|---|---
num. layers | int | 1 | 3
dropout | float | 0.1 | 0.6
embedding | int | 32 | 256
hidden size | int | 64 | 1024
Table 3: Parameter ranges used for hyperparameter tuning.

It turns out that only two of these hyperparameters makes a significant
difference, the number of layers and the hidden size. One LSTM layer was
clearly inferior and so were LSTMs with fewer than 400 neurons per layer. The
embedding dimension and the dropout rate made less difference. We decided to
go with two configurations, a small one with 200 dimensional embedding and the
hidden size set to 256 and a large one with the embedding set to 150
dimensions and 900 hidden size. We report the better one for each language
based on the development set.

###  4.3 Training details

We train the models end-to-end with gradient descent using the Adam optimizer
with 0.001 learning rate. We apply teacher forcing to the decoder with 0.5
probability, which means that we feed the ground truth character instead of
the output of the previous step half of the time.

##  5 Augmentation techniques

In this section we describe the data augmentation techniques we used. We
applied the same steps for each language with varying effect. We performed
ablation studies (Section 7.1) on some languages to investigate the individual
effect of these techniques.

###  5.1 Stem modification

Neural networks tend to have difficulties with low amounts of training data as
is the case with low resource languages. For example a model trained on a
language with 50-150 examples will learn to output the training character
sequences. In order to avoid this we used the data hallucination technique
introduced in Anastasopoulos and Neubig (2019), who identified the “stem”
based on common substrings in the inflected forms of the same lemma. Then they
replace some characters of the stem with random characters. We use a similar
method but instead of using random characters, we sample them according to the
unigram distribution of each language. This way we created 10000 additional
examples for each language in the training set.

###  5.2 Copy

Another attempt was to help the model learn to copy because without it the
model can output wrong characters for the stem instead of copying it. We added
a maximum number of 10000 examples to the training set where the additional
data for each language looks like:
izar izar Lang-family;Lang-code;COPY
Copy is a new tag we added just for this specific task.

Figure 1: The results of the augmentation based on the work of Neuvel and
Fulop (2002), evaluated on underresourced languages. The majority of these
results show that this kind of augmentation is not helpful.

###  5.3 Template based augmentation

We also experimented with a morpheme identifier algorithm of Neuvel and Fulop
(2002). The main idea of Neuvel and Fulop (2002) is to iteratively find
“similar” substrings between inflected forms within the same languages. Common
substrings are considered affixes and they can be used in templates to
generate further examples. We first identify whether the language is more
likely to use prefixes or suffixes. We then apply the forward-comparing or the
backward-comparing method to extract the affixes. The forward-comparing method
is explained in Algorithm 1. The backward-comparing method is identical except
it starts at the end of the string.

Algorithm 1 Forward-comparing method

initialize 3 empty lists: sim, diff1, diff2

for $i\leftarrow 0,\min{\\{len(w1),len(w2)}$} do

if w1[i] equals w2[i] then

$sim\leftarrow w1[i]$

$diff1\leftarrow"."$

$diff2\leftarrow"."$

else

$sim\leftarrow"."$

$diff1\leftarrow w1[i]$

$diff2\leftarrow w1[i]$

end if

end for

$sim\leftarrow\|len(w1)-len(w2)\|$ ”.” characters

Append the rest of the remaining words to the appropriate diff list

return $diff1,diff2,sim$

We store these 3 lists in a single data structure called “comparison”. We
collect comparisons from all the records of our source data. Afterwards we try
to merge them based on their similarities. Two comparisons are considered
similar if in all of their three lists they only differ in ’.’ characters. If
we find such comparisons, the differing ’.’ characters in all lists are
replaced by ’?’ marks - creating a new comparison and the source comparisons
are deleted. This merged comparison then can be considered for further merging
too. We also count how many merges happen in a comparison. The results of this
algorithm can be used to generate newer words by using diff1 and diff2 of a
comparison as templates. Starting from the left side of the template we
generate exactly one character to replace each ’.’ characters (this one
generated character is used to replace the holders in both diff1 and diff2),
and one or zero character to replace each ’?’ marks. We tried improving our
results by generating the characters based on the frequency of the characters
of a given source. We also tried improving under resourced language results by
using words generated by templates from the languages’ families.

The results with and without the template based augmentation are compared in
Figure 1. We only include a handful of low resource languages for clarity.
Template based augmentation is useful for some languages but it decreased the
performance for others and it was computationally expensive to try models with
and without template based augmentation for all languages.

##  6 Experiments

We first experimented with a single large model for all languages. This model
worked reasonably well for large languages and some of the medium ones as
well, however, its performance was quite low on low resourced languages. To
offset this, we introduce a three-step training method.

#### Step 1: General training.

In the first phase, we train a model on all languages with _copy_
augmentation. This step creates a general language model. Using _copy_ alone
considerably improves the results on most languages. We further explore the
effect of _copy_ in Section 7.1. We trained a small and a large model as
described in Section 4.

#### Step 2: Language family training.

We fine-tuned both model for each language family but this time we also added
10000 example to each language with using the _stem modification_
augmentation. This resulted in 24 models, one small and one large for each of
the 12 language families.

#### Step 3: Language specific training.

We then further fine-tune the family models on each language of the particular
family. We use template based augmentation in this step to further increase
the number of training samples.

##  7 Results

Our three step model achieves 81.51% accuracy on average with 0.627 average
Levenshtein distance between the output and the ground truth. This results is
lower than the baseline (84.57% accuracy and 0.448 edit distance) but it is
important to note that our LSTM seq2seq model is simpler than the Transformer
baseline, while it remains competitive. We chose our model based on the
accuracy of the development dataset. This way we submitted the results of 7
smaller and 31 larger models.

###  7.1 Ablation studies

We perform a small ablation studies on two augmentation technique and on each
step of training process. We picked all languages where our results were
better than the baseline and retrained the model without one of the
augmentation techniques or without one of the three training steps. The
results are listed in Table 2. It is clear that _copy_ is the most important
data augmentation technique. _Stem modification_ often has a mixed effect on
these languages. The model on Ludic, the smallest language in the data set, is
actually much better without stem modification. The same is true about the
training steps. The overall accuracy increased with each step but the
language-specific effects are more varied. Since testing with and without all
options is computationally too expensive for such a large number of languages,
we recommend trying these options individually for language-specific
applications.

##  8 Conclusion

We presented the BME team’s submission for the SIGMORPHON 2021 Task 0 Part 1
shared task. We used a standard LSTM encoder-decoder model with attention. We
augmented the data with three different techniques and then used a three-step
training method that first trains on all languages, then fine-tunes on the
language families and finally fine-tunes on the individual languages. Our
submission outperformed the other team by a large margin. Although our results
did not outperform the baseline, we presented a simpler model with multiple
data augmentation techniques that do not require any external resource or
linguistic expertise.

## References

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  * Bergmanis et al. (2017) Toms Bergmanis, Katharina Kann, Hinrich Schütze, and Sharon Goldwater. 2017\.  Training data augmentation for low-resource morphological inflection.  In _Proceedings of the CoNLL SIGMORPHON 2017 Shared Task: Universal Morphological Reinflection_ , pages 31–39, Vancouver. Association for Computational Linguistics.
  * Cotterell et al. (2018) Ryan Cotterell, Christo Kirov, John Sylak-Glassman, Géraldine Walther, Ekaterina Vylomova, Arya D. McCarthy, Katharina Kann, Sabrina J. Mielke, Garrett Nicolai, Miikka Silfverberg, David Yarowsky, Jason Eisner, and Mans Hulden. 2018.  The CoNLL–SIGMORPHON 2018 shared task: Universal morphological reinflection.  In _Proceedings of the CoNLL–SIGMORPHON 2018 Shared Task: Universal Morphological Reinflection_ , pages 1–27, Brussels. Association for Computational Linguistics.
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# On Representations of Affine Temperley–Lieb Algebras

R.M. Green  Mathematical Institute
Oxford University
24–29 St. Giles’
Oxford OX1 3LB
England
E-mail: greenrmaths.ox.ac.uk (now rmgcolorado.edu)

We study the finite-dimensional simple modules, over an algebraically closed
field, of the affine Temperley–Lieb algebra corresponding to the affine Weyl
group of type $A$. These turn out to be closely related to the simple modules
for a certain $q$-analogue of the annular algebra of V.F.R. Jones.

††support: The author was supported in part by an E.P.S.R.C. postdoctoral
research assistantship.

This paper was published in 1998 in the Proceedings of the

8th International Conference on Representations of Algebras

## 1\. Introduction

The Temperley–Lieb algebra is a finite-dimensional algebra which was
introduced in [6] and has been extensively studied in many papers, for example
[7].

In [1], the author and C.K. Fan introduced a diagram calculus for the “affine”
Temperley–Lieb algebra, an infinite-dimensional algebra which is related to
affine Weyl group $W(\widehat{A}_{l})$ of type $A$ in the same way as the
(ordinary) Temperley–Lieb algebras are related to the group algebras of the
symmetric groups. One would like to be able to classify the finite-dimensional
irreducible representations for this algebra, partly because this gives
irreducible representations of the affine Hecke algebra
${\Cal{H}}(\widehat{A}_{l})$ and of the corresponding Weyl group
$W(\widehat{A}_{l})$. The “even rank” case (where $l+1$, the number of nodes
in the extended Dynkin diagram, is even) is already well understood (see [4],
[5]).

The diagram calculus in [1] is reminiscent of the diagram calculus of certain
“annular” algebras which were introduced in [3], and which (following [2]) we
call Jones algebras. It is natural to wonder whether there is a strong
connection between them, especially as both are essentially quotients of the
affine Hecke algebra.

This paper sheds light on these problems and explains how they are related. We
will always work in the category of finite-dimensional modules for the affine
Temperley–Lieb algebra, so the term “irreducible module” should be understood
to mean “finite-dimensional irreducible module”.

After reviewing, in §2, the diagram calculus of [1], we introduce
$q$-analogues of the Jones algebras in §3. (The algebras of [3] give rise to
the case $q=1$.) These $q$-Jones algebras (and certain subalgebras of them)
turn out to be cellular algebras in the sense of [2], in the same way as the
ordinary Jones algebras were cellular (see [2, §6]). This means that we can
easily classify their irreducible modules over an algebraically closed field.
In §4 we explain the connection between the objects of §2 and those of §3.
This means that “most” of the irreducible modules for the affine
Temperley–Lieb algebras are the same as those described in §3, under certain
natural identifications.

This method deals with all the modules except those in the most dominant
“cell” of the algebra in even rank. These do not behave in the same way, and
we do not deal with them here. The reader is referred to [4, §4.3.1], where
these modules are considered.

## 2\. Affine Temperley–Lieb Algebras

The aim of §2 is to review the definition of the affine Temperley–Lieb algebra
as given in [1, §4], and to explain how it is related to certain other
infinite-dimensional algebras given by a calculus of diagrams.

### 2.1 The algebra ${D_{n}}$

It turns out for our purposes to be convenient (for various reasons which will
become clear) to work with an algebra of diagrams which contains the
Temperley–Lieb algebra as a proper subalgebra. We start this section by
defining this bigger algebra.

First, we recall the definition of an affine $n$-diagram from [1, §4.1].

###### Definition 2.1.1

An affine $n$-diagram, where $n\in{\mathbb{Z}}$ satisfies $n\geq 3$, consists
of two infinite horizontal rows of nodes lying at the points
$\\{{\mathbb{Z}}\times\\{0,1\\}\\}$ of ${\mathbb{R}}\times{\mathbb{R}}$,
together with certain curves, called edges, which satisfy the following
conditions:

(i) Every node is the endpoint of exactly one edge.

(ii) Any edge lies within the strip ${\mathbb{R}}\times[0,1]$.

(iii) If an edge does not link two nodes then it is an infinite horizontal
line which does not meet any node. Only finitely many edges are of this type.

(iv) No two edges intersect each other.

(v) An affine $n$-diagram must be invariant under shifting to the left or to
the right by $n$.

We will identify any two diagrams which are isotopic to each other, so that we
are only interested in the equivalence classes of affine $n$-diagrams up to
isotopy. This has the effect that the only information carried by edges which
link two nodes is the pair of vertices given by the endpoints of the edge.
Later on in the paper we will consider “annular involutions” which allow the
diagrams to be described without the use of curves, but the geometric
viewpoint is often helpful.

Another way to describe the equivalence of diagrams is as follows. If $D_{1}$
and $D_{2}$ are two affine $n$-diagrams and there is a bijection $\phi$
between the edges of $D_{1}$ with the edges of $D_{2}$ such that the set of
endpoints of each edge $E\in D_{1}$ (possibly empty) is equal to the set of
endpoints of the edge $\phi(E)$ in $D_{2}$ (possibly empty), then we say that
$D_{1}$ and $D_{2}$ give the same graph, and we identify the diagrams.

Because of the condition (v), one can also think of affine $n$-diagrams as
diagrams on the surface of a cylinder, or within an annulus, in a natural way.
Unless otherwise specified, we shall henceforth regard the diagrams as
diagrams on the surface of a cylinder with $n$ nodes on top and $n$ nodes on
the bottom. Later, we will compare this with Jones’ (finite-dimensional)
annular algebra, which was introduced in [3]. From now on, we will call the
affine $n$-diagrams “diagrams” for short, when the context is clear. Under
this construction, the top row of nodes becomes a circle of $n$ nodes on one
face of the cylinder, which we will refer to as the top circle. Similarly, the
bottom circle of the cylinder is the image of the bottom row of nodes.

The cylindrical viewpoint allows one to see why there is an infinite number of
affine $n$-diagrams: edges which connect nodes on the top circle to nodes on
the bottom circle may wind round the cylinder an arbitrary number of times,
and different winding numbers give nonequivalent diagrams. We note that the
idea of diagrams about a cylinder is familiar from Potts models in statistical
mechanics (see the “boundary diagrams” in [4]).

An example of an affine $n$-diagram for $n=4$ is given in Figure 1. The dotted
lines denote the periodicity, and should be identified to regard the diagram
as inscribed on a cylinder.

Figure 1 An affine 4-diagram

###### Definition 2.1.2

An edge of the diagram $D$ is said to be vertical if it connects a point in
the top circle of the cylinder to a point in the bottom circle, and horizontal
if it connects two points in the same circle of the cylinder.

Two diagrams, $A$ and $B$ “multiply” in the following way, which was described
in [1, §4.2]. Put the cylinder for $A$ on top of the cylinder for $B$ and
identify all the points in the middle row. This produces a certain (natural)
number $x$ of loops. Removal of these loops forms another diagram $C$
satisfying the conditions in Definition 2.1.1. The product $AB$ is then
defined to be $[2]^{x}C$, where $[2]$ is the Laurent polynomial $v+v^{-1}$ in
the indeterminate $v$. It is clear that this defines an associative
multiplication.

###### Definition 2.1.3

Let $R$ be a ring containing an invertible indeterminate $v$. We define the
associative algebra ${D_{n}}$ over $R$ to be the $R$-linear span of all the
affine $n$-diagrams, with multiplication given as above.

### 2.2 The affine Temperley–Lieb algebra

We assume from now on that $n\geq 3$, so there are at least three nodes in
each circle of the cylinder.

###### Definition 2.2.1

Let $R$ be a commutative ring (with identity) containing $[2]$. The affine
Temperley–Lieb algebra ${TL(\widehat{A}_{n-1})}$ is the $R$-algebra given by
generators $E_{1},\ldots,E_{n}$ and defining relations

$\eqalignno{E_{i}^{2}&=[2]E_{i},&(1)\cr E_{i}E_{j}&=E_{j}E_{i},\quad\text{ if
$\bar{i}\neq\overline{j\pm 1}$},&(2)\cr E_{i}E_{\overline{i\pm
1}}E_{i}&=E_{i}.&(3)\cr}$

Here, $\bar{i}$ denotes the congruence class modulo $n$ of $i$.

###### Remark 2.2.2

If $n$ is even, and $R$ is a field of characteristic $0$, this agrees with the
“periodic Temperley–Lieb algebra” studied in [4], although our notation is
different. The generator $U_{i}$ in [4] corresponds to our generator
$E_{2i-1}$, and the generator $U_{ij}$ (where $j=i+1$) in [4] corresponds to
our generator $E_{2i}$.

The algebra ${TL(\widehat{A}_{n-1})}$ occurs as a quotient of the affine Hecke
algebra of type $\widehat{A}$ with algebra generators $T_{1},T_{2},\ldots
T_{n}$. The generator $E_{i}$ is the image of $v^{-1}T_{e}+v^{-1}T_{i}$, where
$T_{e}$ is the identity in the Hecke algebra, and the parameter $q$ in the
Hecke algebra is identified with $v^{2}$.

It was shown in [1, §4] that ${TL(\widehat{A}_{n-1})}$ can also be realised as
an algebra of diagrams, as follows.

###### Proposition 2.2.3

The algebra ${TL(\widehat{A}_{n-1})}$ is the subalgebra of ${D_{n}}$ spanned
by diagrams, $D$, with the following additional properties:

(i) If $D$ has no horizontal edges, then $D$ is the identity diagram, in which
point $j$ in the top circle of the cylinder is connected to point $j$ in the
bottom circle for all $j$.

(ii) If $D$ has at least one horizontal edge, then the number of intersections
of $D$ with the line $x=i+1/2$ for any integer $i$ is an even number.

The identification sends the generator $E_{i}$ to the diagram where there is a
horizontal edge of minimal length connecting $i$ and $\overline{i+1}$ in each
of the circles of the cylinder, and there is a vertical edge connecting $j$ in
the top circle to $j$ in the bottom circle whenever $j\neq i,\overline{i+1}$.

###### Demonstration Proof

This follows from [1, Corollary 4.5.1]. ∎

###### Demonstration Remarks

The number of intersections of $D$ with a line $x=i+1/2$ is, strictly
speaking, not well defined, because edges may “loop back on themselves”. We
define the “number of intersections” as (in (ii)) to be the minimal number
possible, which is obtained by perturbing the diagram of $D$ so that each edge
crosses the line as few times as possible. Precise details may be found in [1,
§4.4].

A typical example of the representation of a generator $E_{i}$ as a diagram on
a cylinder is shown in Figure 2.

Figure 2 The element $E_{2}$ for $n=5$

We now define an important element $u\in{D_{n}}$, similar to the element $u$
of Jones’ annular algebra which was considered in [3].

###### Definition 2.2.4

The diagram $u$ of ${D_{n}}$ is the one satisfying the property that for all
$j\in\text{\bf n}$, the point $j$ in the bottom circle is connected to point
$\overline{j+1}$ in the top circle by a vertical edge taking the shortest
possible route.

In the case $n=4$, the element $u$ is as shown in Figure 3.

Figure 3 The element $u$ for $n=4$

### 2.3 The algebra ${O_{n}}$

We also introduce a subalgebra ${O_{n}}$ of ${D_{n}}$. This is related to
${D_{n}}$ in the same way as the alternating groups are related to the
symmetric groups. (The letter “O” conceptually stands for “oriented”, which is
terminology borrowed from the paper [3].)

Recall from the remarks following Proposition 2.2.3 that the number of
intersections of a diagram $D$ with a line $x=i+1/2$ refers to the minimum
possible such number (since we are allowed to replace $D$ by one of its
isotopic images). In the results which follow, we are usually only interested
in whether this number is even or odd, in which case it is not necessary that
we take the minimum possible number of intersections, provided that any edges
tangent to the line $x=i+1/2$ are not counted as intersections.

###### Definition 2.3.1

Define ${O_{n}}$ to be the $R$-submodule of ${D_{n}}$ spanned by diagrams $D$
such that the number of intersections of $D$ with the line $x=i+1/2$ for any
integer $i$ is an even number.

Such diagrams are said to have the even intersection property. Conversely, if
all the numbers of intersections of a diagram $D$ with the lines $x+1/2$ are
odd, $D$ is said to have the odd intersection property.

###### Lemma 2.3.2

The following identity holds in ${D_{n}}$, regarding the generators $E_{i}$ of
${TL(\widehat{A}_{n-1})}$ as elements of ${D_{n}}$:

$\underbrace{E_{\overline{i+2}}E_{\overline{i+3}}\cdots
E_{\overline{i}}}_{n-1\text{\ terms}}=u^{2}E_{i}.$

###### Demonstration Proof

Since $uE_{i}u^{-1}=E_{\overline{i+1}}$, it suffices to check the case $i=1$.
This follows easily from the diagrammatic viewpoint. ∎

###### Lemma 2.3.3

The $R$-module ${O_{n}}$ is the subalgebra of ${D_{n}}$ generated by
${TL(\widehat{A}_{n-1})}$ and the elements $\\{u^{2m}:m\in{\mathbb{Z}}\\}$.

###### Demonstration Proof

It is not hard to see that if a diagram has no horizontal edges, it must be a
power of $u$, and that the diagram lies in ${O_{n}}$ if and only if this power
of $u$ is even. The fact that there are no more diagrams in ${O_{n}}$ now
follows from Proposition 2.2.3.

To check that these diagrams span a subalgebra, it remains to prove that
$u^{2}E_{i}$ and $u^{-2}E_{i}$ lie in ${O_{n}}$ for each $i$. The case of
$u^{2}E_{i}$ follows from Lemma 2.3.2, and the case of $u^{-2}E_{i}$ follows
by an analogous argument. The assertion now follows. ∎

The following lemma shows that any diagram has the even or odd intersection
property, according as it is an element of ${O_{n}}$ or not.

###### Proposition 2.3.4

Let $D$ be a diagram in ${D_{n}}$. If $D\in{O_{n}}$ (respectively,
$D\not\in{O_{n}})$, then the number of intersections of $D$ with the line
$x=i+1/2$ is an even (respectively, odd) number for each $i$.

###### Demonstration Proof

If $D\in{O_{n}}$, it is clear from Lemma 2.3.3 that $D$ has the even
intersection property, because it is true if $D$ is an even power of $u$ or an
element of ${TL(\widehat{A}_{n-1})}$. The converse, that any diagram with the
even intersection property lies in ${O_{n}}$, is equally clear.

Next, we observe that if $D$ has the even (respectively, odd) intersection
property, then $u.D$ has the odd (repsectively, even) intersection property.
This follows by consideration of the product $u.D$ via diagrams.

By consideration of the fact that each point in the diagram $D$ has exactly
one edge emerging from it, and that this edge can either emerge towards the
left, towards the right, or straight upwards, we see that any diagram $D$ in
${D_{n}}$ has either the odd intersection property or the even intersection
property as mentioned in the statement.

The proof now follows. ∎

###### Lemma 2.3.5

As $O_{n}$-bimodules, ${D_{n}}={O_{n}}\oplus u{O_{n}}={O_{n}}\oplus{O_{n}}u$.

###### Demonstration Proof

We observe from the proof of Proposition 2.3.4 that any diagram of ${D_{n}}$
is either a diagram of ${O_{n}}$ or of the form $u.D$ where $D$ is a diagram
of ${O_{n}}$. We also note that for a diagram $D$, $D\in{O_{n}}\Rightarrow
u^{-1}Du\in{O_{n}}$. The proof follows from these facts. ∎

We now define an automorphism ${\varepsilon}$ of ${D_{n}}$ which fixes
${O_{n}}$.

###### Proposition 2.3.6

The $R$-linear map ${\varepsilon}:{D_{n}}\rightarrow{D_{n}}$ is defined by its
effect on the diagrams $D$ as follows:

${\varepsilon}(D)=\begin{cases}D&\text{ if }D\in{O_{n}};\cr-D&\text{ if
}D\not\in{O_{n}}.\cr\end{cases}$

This map is an automorphism of ${D_{n}}$ whose fixed points are the elements
of ${O_{n}}$.

Thus we have ${\varepsilon}(u)=-u$, ${\varepsilon}(u^{-1})=-u^{-1}$ and
${\varepsilon}(E_{i})=E_{i}$ for all $i$.

###### Demonstration Proof

From Lemma 2.3.5, we find that any diagram in ${D_{n}}\backslash{O_{n}}$ can
be written (uniquely) in the form $u.E$ where $E$ is a diagram of ${O_{n}}$.
Conversely, elements of these forms do not lie in ${O_{n}}$. Using the fact
that $u.{O_{n}}.u^{-1}={O_{n}}$ and the fact that even powers of $u$ lie in
${O_{n}}$, we see that the map ${\varepsilon}$ as described respects the
multiplication (because two diagrams in ${D_{n}}\backslash{O_{n}}$ multiply to
give an element of ${O_{n}}$, and so on).

The effect of ${\varepsilon}$ on the generators and the assertion about the
fixed points of ${\varepsilon}$ are easy to prove. ∎

To conclude this section, we note that ${D_{n}}$ has a nice presentation in
terms of generators and relations, as follows.

###### Proposition 2.3.7

The algebra ${D_{n}}$ is generated by elements

$E_{1},\ldots,E_{n},u,u^{-1}.$

It is subject to the relations (1), (2) and (3) of Definition 2.2.1, and the
following additional defining relations:

$\eqalignno{uE_{i}u^{-1}&=E_{\overline{i+1}},&(4)\cr(uE_{1})^{n-1}&=u^{n}.(uE_{1}).&(5)\cr}$

###### Demonstration Proof

From Lemma 2.3.3 and Lemma 2.3.5, we see that the given set does generate the
algebra ${D_{n}}$. In addition, all the relations are true; (5) looks
unfamiliar but we claim that it is equivalent to Lemma 2.3.2 after
applications of (4). To do this, first reexpress (5) as
$u^{-(n-1)}(uE_{1})^{n-1}=u^{2}E_{1}$. Now apply (4) to quasi-commute all the
occurrences $u$ in the left hand side to the left of the expression. This
establishes the special case of Lemma 2.3.2 where $i=1$. To obtain the general
expression, we conjugate each side of the equation by a suitable power of $u$.

We then observe that any word in the generators can be rewritten as a multiple
of a power of $u$ times a monomial in the $E_{i}$ (i.e. a diagram of
${TL(\widehat{A}_{n-1})}$) by using the relations (1) to (4). Using Lemma
2.3.2 (i.e. relation (5)), one can reduce this power of $u$ to $0$ or $1$,
unless the word is simply a power of $u$. Lemma 2.3.5 now shows that the
relations are sufficient. ∎

###### Remark 2.3.8

In fact, ${D_{n}}$ is generated by $u$, $u^{-1}$ and $E_{1}$, as can be seen
from relation (4).

## 3\. $q$-Jones Algebras

In §3, we assume that $K=F(v)$ is an algebraically closed field, where $F$ is
a subfield of $K$ and $v$ is an indeterminate. We then introduce a family of
finite-dimensional algebras $J_{q}(n)$, where $q$ is a nonzero element of $K$.
We call $J_{q}(n)$ the $q$-Jones algebra of rank $n$, where $n$ is a natural
number at least $3$. The behaviour of these algebras in odd and in even ranks
turns out to be distinctly different; if $n$ is odd then $J_{1}(n)$ is Jones’s
annular algebra, $J(n)$, as defined in [3]. However, if $n$ is even,
$J_{1}(n)$ is not the full Jones algebra but a quotient of the subalgebra
spanned by the “oriented elements” of $J(n)$. In both even and odd case, the
algebras turn out to be cellular in the sense of [2], which enables their
simple modules over $K$ to be easily determined. These $q$-Jones algebras
appear to be new.

### 3.1 Cellular algebras

For completeness, we recall the definition of a cellular algebra from [2].

###### Definition 3.1.1

Let $R$ be a commutative ring with identity. A cellular algebra over $R$ is an
associative unital algebra, $A$, together with a cell datum $(\Lambda,M,C,*)$
where

1\. $\Lambda$ is a poset. For each ${\lambda}\in\Lambda$, $M({\lambda})$ is a
finite set (the set of “tableaux” of type ${\lambda}$) such that

$C:\coprod_{{\lambda}\in\Lambda}\left(M({\lambda})\times
M({\lambda})\right)\rightarrow A$

is injective with image an $R$-basis of $A$.

2\. If ${\lambda}\in\Lambda$ and $S,T\in M({\lambda})$, we write
$C(S,T)=C_{S,T}^{{\lambda}}\in A$. Then $*$ is an $R$-linear involutary anti-
automorphism of $A$ such that $(C_{S,T}^{{\lambda}})^{*}=C_{T,S}^{{\lambda}}$.

3\. If ${\lambda}\in\Lambda$ and $S,T\in M({\lambda})$ then for all $a\in A$
we have

$a.C_{S,T}^{{\lambda}}\equiv\sum_{S^{\prime}\in
M({\lambda})}r_{a}(S^{\prime},S)C_{S^{\prime},T}^{{\lambda}}\mod
A(<{\lambda}),$

where $r_{a}(S^{\prime},S)\in R$ is independent of $T$ and $A(<{\lambda})$ is
the $R$-submodule of $A$ generated by the set

$\\{C_{S^{\prime\prime},T^{\prime\prime}}^{\mu}:\mu<{\lambda},S^{\prime\prime}\in
M(\mu),T^{\prime\prime}\in M(\mu)\\}.$

### 3.2 Annular Involutions

Let $D$ be a diagram (as in §2) representing a basis element of the extended
affine Temperley–Lieb algebra ${D_{n}}$. Throughout §3.2, we are only
concerned with diagrams $D$ with at least one vertical edge. We define a
certain integer $w(D)$ as follows.

###### Definition 3.2.1

Let $D$ be as above. Let $w_{1}(D)$ be the number of pairs
$(i,j)\in{\mathbb{Z}}\times{\mathbb{Z}}$ where $i>j$ and $\overline{j}$ in the
bottom circle of $D$ is joined to $\overline{i}$ in the top circle of $D$ by
an edge which crosses the “seam” $x=1/2$. We then define $w_{2}(D)$ similarly
but with the condition that $i<j$, and we define $w(D)=w_{1}(D)-w_{2}(D)$.

###### Remark 3.2.2

We think of $w(D)$ as measuring some kind of “winding number” of the diagram
$D$.

The non-intersection condition guarantees that at least one of $w_{1}(D)$ or
$w_{2}(D)$ is $0$, and that $w(D)$ is always finite. However, the values of
$w(D)$ for $D\in{D_{n}}$ are unbounded even though $n$ is fixed.

Some examples of the values of $w(D)$ are as follows. In Figure 1, the pairs
which cross the seam are $(3,-1)$ and $(4,0)$, giving $w(D)=2$. Figure 2 has
$w(D)=0$. Figure 3 has $w(D)=1$, corresponding to the pair $(1,0)$. A moment’s
thought shows that $w(u^{n})=n$ for any $n\in{\mathbb{Z}}$, which motivates
the definition.

We recall the definition of an annular involution of the symmetric group from
[2, Lemma 6.2].

###### Definition 3.2.3

An involution $S\in{\Cal{S}}_{n}$ is annular if and only if, for each pair
$i,j$ interchanged by $S$ $(i<j)$, we have

(a) $S[i,j]=[i,j]$ and

(b) $[i,j]\cap\text{\rm Fix}S=\emptyset$ or $\text{\rm Fix}S\subseteq[i,j]$.

We write $S\in I(t)$ if $S$ has $t$ fixed points, and we write $S\in\text{\rm
Ann({\bf n})}$ if $S$ is annular.

###### Lemma 3.2.4

The diagrams of ${D_{n}}$ with vertical edges are in canonical bijection with
triples $[S_{1},S_{2},w]$ where $S_{1},S_{2}\in\text{\rm Ann({\bf n})}\cap
I(t)$ for $t>0$, and $w\in{\mathbb{Z}}$.

###### Demonstration Proof

The correspondence is given as follows. Consider a diagram $D$. The involution
$S_{1}$ exchanges points $i$ and $j$ if and only if $i$ is connected to $j$ in
the top circle of $D$. Similarly $S_{2}$ exchanges points $i$ and $j$ if and
only if $i$ is connected to $j$ in the bottom circle of $D$. One can easily
see that $S_{1}$ and $S_{2}$ are annular. Finally $w=w(D)$; this determines
which points are connected by the vertical edges (it is enough to specify one
vertical edge, by the non-intersection criterion).

Conversely consider a triple $[S_{1},S_{2},w]$ as required. The hypothesis
$t>0$ guarantees that each of $S_{1}$ and $S_{2}$ each has a fixed point. If
$S\in{S_{1},S_{2}}$ exchanges points $i$ and $j$, it now follows from part (b)
of the definition that we can determine which way round the cylinder the
corresponding edge in the diagram goes. Using part (a) we can show that no
intersections occur, so $S_{1}$ determines the horizontal edges in the top of
the diagram and $S_{2}$ determines those in the bottom. As regards $w$,
observe that one can disconnect all the vertical edges from a diagram and
reconnect the top end of each one to the next available point to the right.
This increases $w(D)$ by 1. Conversely one can decrease it by one. Thus $w$
determines the endpoints of all the vertical edges, which completes the proof.
∎

###### Definition 3.2.5

For $S_{1},S_{2}\in\text{\rm Ann({\bf n})}\cap I(t)$ where $t>0$, we define
$r(S_{1},S_{2})$ to be the smallest nonnegative integer (which will be $0$ or
$1$, following Proposition 2.3.4) satisfying
$[S_{1},S_{2},r(S_{1},S_{2})]\in{O_{n}}$.

### 3.3 $q$-Jones Algebras in Odd Rank

In §3.3, we assume that $n$ is an odd number. (We deal with this before the
even case because the even case is already better understood.)

We now put an equivalence relation, $\sim$, on the basis elements
$[S_{1},S_{2},w]$.

###### Definition 3.3.1

We say that $[S_{1},S_{2},w]\sim[T_{1},T_{2},w^{\prime}]$ if and only if
$S_{1}=T_{1}$, $S_{2}=T_{2}$ and $w$ and $w^{\prime}$ are congruent modulo
$t$, where $S_{1},S_{2}\in\text{\rm Ann({\bf n})}\cap I(t)$.

We say a basis element $[S_{1},S_{2},w]$ is in root position if and only if
$0\leq w<t$. Clearly the set of basis elements in root position is a set of
equivalence class representatives for $\sim$. We usually denote such an
element by $[S_{1},S_{2},{\tau}_{i}]$ to emphasize that it is in root
position.

###### Lemma 3.3.2

The subspace spanned by elements of the form

$\\{[S_{1},S_{2},w]-q^{s}[T_{1},T_{2},w^{\prime}]:[S_{1},S_{2},w]\sim[T_{1},T_{2},w^{\prime}]\text{\rm\
and \ }w^{\prime}=w+ts\\},$

where $S_{1}\in\text{\rm Ann({\bf n})}\cap I(t)$, is an ideal of ${D_{n}}$. We
denote it by ${\omega}(q)$.

###### Demonstration Proof

It is clear from the diagram calculus that the product of any two basis
elements of ${D_{n}}$ is a multiple of another one. If

$[X_{1},X_{2},w][Y_{1},Y_{2},w^{\prime}]={\beta}[Z_{1},Z_{2},w^{\prime\prime}]$

for some scalar ${\beta}$ then one can also easily see that

$[X_{1},X_{2},w][Y_{1},Y_{2},w^{\prime}+t]={\beta}[Z_{1},Z_{2},w^{\prime\prime}+t].$

The claim now follows. ∎

###### Definition 3.3.3

The $q$-Jones algebra over $K(q)$ (where $q$ is an indeterminate) is the
quotient of ${D_{n}}$ by ${\omega}(q)$.

###### Remark 3.3.4

When $q=1$, one recovers the usual Jones algebra (see [2, Proposition 6.14]).

###### Lemma 3.3.5

The $q$-Jones algebra has a basis given by the images of the elements

$\\{[S_{1},S_{2},{\tau}_{i}]:S_{1},S_{2}\in\text{\rm Ann({\bf n})}\cap
I(t),{\tau}_{i}\in\\{0,1,\ldots,t-1\\}\\}$

###### Demonstration Proof

It is clear that the given elements span the algebra.

Suppose for a contradiction that there is some nontrivial linear relation
between the elements. Then we may assume that the coefficients of this
relation lie in $K[q]$ and that not all of them have $(q-1)$ as a factor.
Putting $q=1$ and using Remark 3.3.4, we now obtain a nontrivial linear
relation for the corresponding basis elements of the Jones algebra, which is a
contradiction. ∎

### 3.4 Representation theory of the $q$-Jones algebra in odd rank

We can now prove that the $q$-Jones algebra is cellular. The motivation behind
doing this is that we will then have bases for all its finite dimensional
irreducible modules over $K$. The methods are the same as those in [2, §6].

We specialize the indeterminate $q$ to a fixed, nonzero element of $K$. We
also define the set ${\Cal{T}}(n)$ to be the set of odd natural numbers up to
and including $n$. We abuse notation by identifying an element
$[S_{1},S_{2},w]$ where $0\leq w<t$ with its image in $J_{q}(n)$.

###### Definition 3.4.1

Let $w_{0}$ be the longest element of ${\Cal{S}}_{n}$. If $S\in\text{\rm
Ann({\bf n})}$, we define $S^{*}\in\text{\rm Ann({\bf n})}$ to be the element
$w_{0}^{-1}Sw_{0}$.

###### Proposition 3.4.2

Let $t\in{\Cal{T}}(n)$. We factorise

$x^{t}-q=\prod_{i=1}^{t}(x-r_{i}(t,q))$

and write

$f_{j}(x,q)=\prod_{i>j}(x-r_{i}(t,q))=\sum_{i=0}^{t-j}r_{ij}(t,q)x^{i}$

for $j=1,\ldots,t$. Write $f_{0}(x,q)=1$ so that $r_{00}(0)=1$. Then a cell
datum for $J_{q}(n)$ over $K$ is given by $(\Lambda,M,C,*)$ defined as
follows:

(a) $\Lambda=\\{(t,j):t\in{\Cal{T}}(n),j\in\text{\bf t}\\}$, ordered
lexicographically, i.e. $(t^{\prime},j^{\prime})<(t,j)$ if and only if
$t^{\prime}<t$ or $t=t^{\prime},j^{\prime}<j$.

(b) For $(t,j)\in\Lambda$, $M(t,j)=\text{\rm Ann({\bf n})}\cap I(t)$.

(c) If $S_{1},S_{2}\in M(t,j)$ then
$C_{S_{1},S_{2}}^{t,j}=\sum_{i=0}^{t-j}r_{ij}(t,q)[S_{1},S_{2}^{*},i].$

(d) The anti-automorphism, $*$, is defined by
$[S_{1},S_{2},w]^{*}=[S_{2}^{*},S_{1}^{*},w]$.

###### Demonstration Proof

We check that the three conditions of Definition 3.1.1 hold. Condition 1 comes
from Lemma 3.3.5.

To prove condition 2, we observe that applying $*$ to a basis element
$[S_{1},S_{2}^{*},w]$ in root position gives another element in root position.
The fact that the map is an anti-automorphism is clear from the diagrammatic
viewpoint—the cylinders are being rotated so they are upside-down.

The proof of condition 3 is the same as [2, Theorem 6.15], with a few minor
and obvious changes. ∎

The motivation behind proving Proposition 3.4.2 is that the theory of cellular
algebras allows one to construct a complete list of finite-dimensional
irreducible modules for $J_{q}(n)$.

###### Definition 3.4.3

Let ${\lambda}=(t,j)\in\Lambda$. The module $W(t,j,q)$ for $J_{q}(n)$ has a
basis given by $\\{C_{S}:S\in M(t,j)\\}$ with action specified as follows. If
$a\in J_{q}(n)$ and

$a.C_{S,T}^{t,j}\equiv\sum_{S^{\prime}\in
M(t,j)}c_{a}(S^{\prime},S)C_{S^{\prime},T}^{t,j}\mod J_{q}(n)\ (<{\lambda})$

for any suitable $T$ (for example, $S$), then the left action of $a$ on
$W(t,j,q)$ is defined by

$a.C_{S}=\sum_{S^{\prime}\in M(t,j)}c_{a}(S^{\prime},S)C_{S^{\prime}}.$

###### Remark 3.4.4

The theory of cellular algebras in [2] ensures that this is well-defined.

###### Proposition 3.4.5

The modules $W(t,j,q)$, as $r_{j}(t,q)$ ranges over the $t$-th roots of $q$,
are a complete set of irreducible representations for $J_{q}(n)$ over $K$.

###### Demonstration Proof

This is essentially a restatement of [2, Corollary 6.17], after making the
necessary trivial changes to [2, Scholium 6.16]. Note that the parameter
$\delta$ occurring in [2, Corollary 6.17] is equal to $[2]=v+v^{-1}\neq 0$ in
our case. ∎

###### Proposition 3.4.6

Let $\psi$ be a representation corresponding to the irreducible module
$W(t,j,q)$, and let $[S_{1},S_{2},w]$ be a basis element of $J_{q}(n)$ arising
from the set $M(t,j)$. Then

$\psi([S_{1},S_{2},w+1])=r_{j}(t,q).\psi([S_{1},S_{2},w]).$

###### Demonstration Proof

This is a corollary of the proof of Proposition 3.4.2. For more details see
[2, (6.15.1), (6.15.2)]. ∎

###### Remark 3.4.7

Proposition 3.4.6 gives a convenient way to test whether the representations
$W(t,j,q)$ and $W(t,j^{\prime},q)$ are distinct. We will use this fact later
in the classification of the simple modules for ${TL(\widehat{A}_{n-1})}$.

### 3.5 $q$-Jones Algebras in Even Rank

We now highlight the differences between the odd rank situation and the
corresponding even rank situation. We treat the case of even rank in less
detail than the case of odd rank studied in §3.3 and §3.4, because the
situation is already better understood and the techniques are very similar.

In §3.5, $n$ is even and the algebra ${O_{n}}$ takes over the role which
${D_{n}}$ played in §3.3 and §3.4. Since $n$ is even, any diagram has an even
number of vertical edges, and so the relation $\sim$ of Definition 3.3.1
carries over naturally to an equivalence relation on the basis elements of
${O_{n}}$.

For $n$ even, we set ${\Cal{T}}(n):=\\{0,2,4,\ldots,n\\}$.

###### Lemma 3.5.1

The diagrams of ${O_{n}}$ with vertical edges are in canonical bijection with
triples $[S_{1},S_{2},r(S_{1},S_{2})+w]$ where
$t\in{\Cal{T}}(n)\backslash\\{0\\}$, $S_{1},S_{2}\in\text{\rm Ann({\bf
n})}\cap I(t)$, and $w\in 2{\mathbb{Z}}$.

###### Demonstration Proof

This is along the same lines as the proof of Lemma 3.2.4. ∎

###### Remark 3.5.2

The hypothesis that there exists a vertical edge in the diagram is necessary,
otherwise one could form a counterexample using any diagram with no vertical
edges!

###### Definition 3.5.3

The $R$-module ${\omega}^{\prime}(q)$ of ${O_{n}}$ is the span of elements of
${O_{n}}$ of the form given in Lemma 3.3.2 where $[S_{1},S_{2},w]$, and hence
$[T_{1},T_{2},w^{\prime}]$, involve vertical edges. The ideal $I_{0}$ of
${O_{n}}$ is the span of the elements of ${O_{n}}$ with no vertical edges.

The $q$-Jones algebra over $K(q)$ (where $q$ is an indeterminate) is the
quotient of ${O_{n}}$ by the ideal ${\omega}^{\prime}(q)+I_{0}(q)$.

###### Remark 3.5.4

It is clear from the diagrammatic viewpoint that $I_{0}(q)$ is an ideal. One
proves that ${\omega}^{\prime}(q)+I_{0}(q)$ is an ideal by using an argument
identical to that of the proof of Lemma 3.3.2.

It should be noted that the term “$q$-Jones algebra” is misleading in even
rank, because specializing $q$ to $1$ gives a smaller algebra than Jones’
annular algebra. In fact, we obtain the quotient of the Jones algebra by its
ideal $J_{0}(n)$ spanned by basis elements with no through-strings. The reason
for factoring out $I_{0}$ is that irreducible modules $M$ satisfying
$I_{0}.M=M$ behave rather differently from those satisfying $I_{0}.M=0$. In
this paper we are interested in the latter class of modules.

###### Lemma 3.5.5

The $q$-Jones algebra $J_{q}(n)$, for even $n$, has a basis given by the
images of the elements

$\\{[S_{1},S_{2},r(S_{1},S_{2})+2w]\\},$

where $t\in{\Cal{T}}(n)\backslash\\{0\\}$, $S_{1},S_{2}\in\text{\rm Ann({\bf
n})}\cap I(t)$ and $w\in\\{0,1,\ldots,t/2-1\\}.$

###### Demonstration Proof

As in Lemma 3.3.5, it is clear that the given elements span the algebra.

Suppose for a contradiction that there is some nontrivial linear relation
between the elements. Then we may assume that the coefficients of this
relation lie in $K[q]$ and that not all of them have $(q-1)$ as a factor.
Putting $q=1$, we now obtain a nontrivial linear relation for the
corresponding basis elements of $J(n)/J_{0}(n)$, and thus a nontrivial linear
relation in $J(n)$, which is a contradiction. ∎

We will abuse notation again by writing $[S_{1},S_{2},w]$ (one of the elements
appearing in Lemma 3.5.5) for its image in $J_{q}(n)$.

We now give a cell datum for $J_{q}(n)$ in even rank.

###### Proposition 3.5.6

Define ${\Cal{T}}^{\prime}(n):=\\{1,2,\ldots,n/2\\}$, and let
$t\in{\Cal{T}}^{\prime}(n)$. We factorise

$x^{t}-q=\prod_{i=1}^{t}(x-r_{i}(t,q))$

and write

$f_{j}(x,q)=\prod_{i>j}(x-r_{i}(t,q))=\sum_{i=0}^{t-j}r_{ij}(t,q)x^{i}$

for $j=1,\ldots,t$. Write $f_{0}(x,q)=1$ so that $r_{00}(0)=1$. Then a cell
datum for $J_{q}(n)$ over $K$ is given by $(\Lambda,M,C,*)$ defined as
follows:

(a) $\Lambda=\\{(t,j):t\in{\Cal{T}}^{\prime}(n),j\in\text{\bf t}\\}$, ordered
lexicographically, i.e. $(t^{\prime},j^{\prime})<(t,j)$ if and only if
$t^{\prime}<t$ or $t=t^{\prime},j^{\prime}<j$.

(b) For $(t,j)\in\Lambda$, $M(t,j)=\text{\rm Ann({\bf n})}\cap I(2t)$.

(c) If $S_{1},S_{2}\in M(t,j)$ then
$C_{S_{1},S_{2}}^{t,j}=\sum_{i=0}^{t-j}r_{ij}(t,q)[S_{1},S_{2}^{*},r(S_{1},S_{2}^{*})+2i].$

(d) The anti-automorphism, $*$, is defined by
$[S_{1},S_{2},w]^{*}=[S_{2}^{*},S_{1}^{*},w]$.

###### Demonstration Proof

This uses exactly the same techniques as Proposition 3.4.2. ∎

###### Remark 3.5.7

Using this cell datum, one can state and prove results exactly analogous to
Definition 3.4.3, Remark 3.4.4, Proposition 3.4.5, Proposition 3.4.6 and
Remark 3.4.7. The only appreciable differences are that the role of
${\Cal{T}}(n)$ in the odd rank is played by ${\Cal{T}}^{\prime}(n)$ in the
even rank, and that the role of $t$ in the odd rank is played by $t/2$ in the
even rank.

## 4\. Classification of Simple Modules for ${TL(\widehat{A}_{n-1})}$

To classify the simple modules for ${TL(\widehat{A}_{n-1})}$, we find once
again that it is helpful to consider the cases of odd and even $n$ separately.
We tackle the case of odd rank first, because it is less well understood.

### 4.1 Simple modules in odd rank

The technique in odd rank is to calculate the simple modules for ${D_{n}}$ and
study their behaviour upon restriction to ${O_{n}}$ and then to
${TL(\widehat{A}_{n-1})}$. We also recall the automorphism ${\varepsilon}$
from §2.

###### Proposition 4.1.1

The irreducible representations $\psi$ for ${D_{n}}$ are parametrised by pairs
$({\alpha}_{t},t)$, where $t\in{\Cal{T}}(n)$ and ${\alpha}_{t}\in K\backslash
0$. The scalar ${\alpha}_{t}$ is that appearing in the equation

$\psi([S_{1},S_{2},w+1])={\alpha}_{t}.\psi([S_{1},S_{2},w]),$

where $S_{1},S_{2}\in\text{\rm Ann({\bf n})}\cap I(t)$.

###### Demonstration Proof

Let $M$ be a simple module for ${D_{n}}$. Let $I_{t}$ be the ideal of
${D_{n}}$ spanned by all diagrams with at most $t$ vertical edges. Then there
exists a unique (odd) number $t$ such that $I_{t}.M=M$ but $I_{t-2}.M=0$,
interpreting $I_{-1}=0$. We also see that $M$ is an irreducible module for
$D_{n}/{I_{t-2}}$. Clearly, $t\in{\Cal{T}}(n)$.

One sees from the diagrammatic viewpoint that $u^{n}$ lies in the centre of
${D_{n}}$. By Schur’s Lemma, $u^{n}$ acts as a scalar, ${\alpha}$, on $M$. The
ideal generated by $u^{n}-{\alpha}$ is the same as the ideal ${\omega}(q)$ in
Definition 3.3.3, for $q={\alpha}$, so $M$ is a simple module for the
$q$-Jones algebra $J_{q}(n)$ for this value of $q$. Proposition 3.4.5 now
shows that the irreducibles are parametrised by the set of $t$-th roots of
${\alpha}$. (Note that we allow the possibility that ${\alpha}$ does not have
$t$ distinct $t$-th roots and there are fewer than $t$ irreducibles.) The fact
that nonisomorphic irreducibles have different parameters comes from
Proposition 3.4.6 and Remark 3.4.7. Conversely, any nonzero ${\alpha}\in K$
turns up as one of the numbers $r_{j}(t,q)$ for $q={\alpha}^{t}$.

This means that all the irreducibles for ${D_{n}}$ are parametrised by pairs
$({\alpha}_{t},t)$, where $t$ is as above and ${\alpha}_{t}\in K\backslash 0$.
The last assertion follows from Proposition 3.4.6. ∎

###### Proposition 4.1.2

The irreducible representations $\rho$ for ${O_{n}}$ are parametrised by pairs
$({\alpha}_{t},t)$, where $t\in{\Cal{T}}(n)$ and ${\alpha}_{t}\in K\backslash
0$. The scalar ${\alpha}_{t}$ is that appearing in the equation

$\rho([S_{1},S_{2},w+2])={\alpha}_{t}.\rho([S_{1},S_{2},w]),$

where $S_{1},S_{2}\in\text{\rm Ann({\bf n})}\cap I(t)$.

###### Demonstration Proof

Let $L$ be a simple module for ${D_{n}}$, and let $L^{\prime}$ be a simple
${O_{n}}$-submodule of $L$ (using the ${O_{n}}$-module structure suggested by
Lemma 2.3.5). Since $u^{n}$ acts as a scalar on $L$, it acts as a scalar on
$L^{\prime}$, and in particular $u^{n}L^{\prime}=L^{\prime}$. However,
$u^{2}L^{\prime}=L^{\prime}$ since $u^{2}\in{O_{n}}$. Since $n$ is odd and $u$
is invertible, this implies $u.L^{\prime}=L^{\prime}$, and thus
$L=L^{\prime}$. Furthermore, $L$ is determined by $L^{\prime}$ and the scalar
by which $u^{n}$ acts.

It is clear that two irreducible modules $L_{1}$ and $L_{2}$ for ${D_{n}}$
restrict to nonisomorphic modules for ${O_{n}}$ unless their parameters
$({\alpha}_{1},t_{1})$ and $({\alpha}_{2},t_{2})$ satisfy $t_{1}=t_{2}$ and
${\alpha}_{1}=\pm{\alpha}_{2}$. (This is because

$\psi([S_{1},S_{2},w+2])={\alpha}^{2}.\psi([S_{1},S_{2},w]),$

where $\psi$ is a representation of ${D_{n}}$ corresponding to a pair
$({\alpha},t)$ and $S_{1},S_{2}\in\text{\rm Ann({\bf n})}\cap I(t)$.) If
$\text{\rm char}K=2$ then necessarily $L_{1}=L_{2}$. Otherwise, let $\psi_{1}$
be a representation affording $L_{1}$. We claim that if $L_{1}$ is given by
the pair $({\alpha}_{t},t)$, then $\psi_{1}.{\varepsilon}$ is an irreducible
representation for ${D_{n}}$ corresponding to the pair $(-{\alpha}_{t},t)$ and
thus affording $L_{2}\neq L_{1}$. This is because

$\psi([S_{1},S_{2},w+1])={\alpha}.\psi([S_{1},S_{2},w])\Rightarrow\psi({\varepsilon}([S_{1},S_{2},w+1]))=-{\alpha}.\psi({\varepsilon}([S_{1},S_{2},w])).$

Irreducibility follows from the fact that $L_{1}$ and $L_{2}$ have the same
dimension. Thus $\psi_{1}.{\varepsilon}$ affords $L_{2}$, although $\psi_{1}$
and $\psi_{1}.{\varepsilon}$ give the same representation of ${O_{n}}$, since
${\varepsilon}$ fixes ${O_{n}}$.

The result now follows from these observations and Frobenius reciprocity. ∎

###### Theorem 4.1.3

The nontrivial irreducible representations $\rho$ for
${TL(\widehat{A}_{n-1})}$ are parametrised by pairs $({\alpha}_{t},t)$, where
$t\in{\Cal{T}}(n)\backslash\\{n\\}$ and ${\alpha}_{t}\in K\backslash 0$. The
scalar ${\alpha}_{t}$ is that appearing in the equation

$\rho([S_{1},S_{2},w+2])={\alpha}_{t}.\rho([S_{1},S_{2},w]),$

where $S_{1},S_{2}\in\text{\rm Ann({\bf n})}\cap I(t)$. The only other
irreducible representation is the “trivial” representation of dimension $1$
which sends the identity to $1$ and all the generators $E_{i}$ to $0$.

###### Demonstration Proof

Recall that ${TL(\widehat{A}_{n-1})}$ and ${O_{n}}$ are filtered by two-sided
ideals indexed by the number of vertical edges in the diagrams. Apart from the
top section, which contains only the identity in ${TL(\widehat{A}_{n-1})}$ but
is infinite in ${O_{n}}$, the sections of this filtration by ideals are the
same (in fact, the ideals themselves are the same). It follows that the simple
modules are the same, except for those corresponding to the top section. In
the latter case, only the “trivial” module (as described in the statement of
the Theorem) for ${TL(\widehat{A}_{n-1})}$ survives. ∎

### 4.2 Simple modules in even rank

Similar techniques can also be used to classify the simple modules for
${TL(\widehat{A}_{n-1})}$ where $n$ is even. As usual we exclude the case
where $I_{0}.M=M$ and $I_{0}$ is as in Definition 3.5.3.

###### Proposition 4.2.1

The irreducible representations $\rho$ for ${O_{n}}$ satisfying
$\rho(I_{0})=0$ are parametrised by pairs $({\alpha}_{t},t)$, where
$t\in{\Cal{T}}^{\prime}(n)$ and ${\alpha}_{t}\in K\backslash 0$. The scalar
${\alpha}_{t}$ is that appearing in the equation

$\rho([S_{1},S_{2},w+2])={\alpha}_{t}.\rho([S_{1},S_{2},w]),$

where $S_{1},S_{2}\in\text{\rm Ann({\bf n})}\cap I(2t)$.

###### Demonstration Proof

Let $M$ be the simple module corresponding to $\rho$. Again, $u^{n}$ acts as a
scalar, ${\alpha}$, on $M$, by Schur’s lemma. Thus the ideal generated by
$u^{n}-{\alpha}$ factors through $M$, and we obtain an irreducible
representation for $J_{q}(n)$ with $q={\alpha}$. The list of such
representations is indexed by the $t$-th roots of ${\alpha}$, and they are
distinguishable as in the statement of the proposition by the usual
techniques. As in Proposition 4.1.1, such a representation exists for any
${\alpha}_{t}$, which completes the proof. ∎

Finally we rederive the result in [4] which parametrises the simple modules
for ${TL(\widehat{A}_{n-1})}$ in even rank.

###### Theorem 4.2.2

The nontrivial irreducible representations $\rho$ for
${TL(\widehat{A}_{n-1})}$ which satisfy $\rho(I_{0})=0$ are parametrised by
pairs $({\alpha}_{t},t)$, where $t\in{\Cal{T}}^{\prime}(n)\backslash\\{n/2\\}$
and ${\alpha}_{t}\in K\backslash 0$. The scalar ${\alpha}_{t}$ is that
appearing in the equation

$\rho([S_{1},S_{2},w+2])={\alpha}_{t}.\rho([S_{1},S_{2},w]),$

where $S_{1},S_{2}\in\text{\rm Ann({\bf n})}\cap I(2t)$. The only other
irreducible representation is the “trivial” representation of dimension $1$
which sends the identity to $1$ and all the generators $E_{i}$ to $0$.

###### Demonstration Proof

This follows the same lines as Theorem 4.1.3, except that it uses Proposition
4.2.1 instead of Proposition 4.1.2. ∎

## Acknowledgements

The author would like to thank C.K. Fan for some helpful discussions. The
author is also grateful to K. Erdmann and the referee for numerous helpful
comments on the manuscript.

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