# Symmetric Petz-Rényi relative entropy uncertainty relation

Domingos S. P. Salazar  Unidade de Educação a Distância e Tecnologia,
Universidade Federal Rural de Pernambuco, 52171-900 Recife, Pernambuco, Brazil

(August 28, 2024)

###### Abstract

Holevo introduced a fidelity between quantum states that is symmetric and as
effective as the trace norm in evaluating their similarity. This fidelity is
bounded by a function of the trace norm, a relationship to which we will refer
as Holevo’s inequality. More broadly, Holevo’s fidelity is part of a one-
parameter family of symmetric Petz-Rényi relative entropies, which in turn
satisfy a Pinsker’s-like inequality with respect to the trace norm. Although
Holevo’s inequality is tight, Pinsker’s inequality is loose for this family.
We show that the symmetric Petz-Rényi relative entropies satisfy a tight
inequality with respect to the trace norm, improving Pinsker’s and reproducing
Holevo’s as a specific case. Additionally, we show how this result emerges
from a symmetric Petz-Rényi uncertainty relation, a result that encompasses
several relations in quantum and stochastic thermodynamics.

_Introduction -_ Holevo introduced a quantum fidelity Holevo (1972) between
states $\rho$ and $\sigma$ defined as

$F_{H}(\rho,\sigma):=[\text{tr}(\sqrt{\rho}\sqrt{\sigma})]^{2},$ (1)

which is symmetric, $F_{H}(\rho,\sigma)=F_{H}(\sigma,\rho)$, and used under
different names in the literature such as affinity and overlap Wilde (2018);
Iten  _et al._ (2017); Ma  _et al._ (2008); Albrecht (1994); Luo and Zhang
(2004); Kim (2013); Audenaert (2014). We have the following inequality for
Holevo’s fidelity,

$T(\rho,\sigma):=\frac{1}{2}|\rho-\sigma|_{1}\leq\sqrt{1-F_{H}(\rho,\sigma)},$
(2)

where $|x|_{1}:=\text{tr}(\sqrt{x^{\dagger}x})$ is the trace norm. More
generally, Holevo’s fidelity is a member of a one parameter family of
symmetric Petz-Rényi relative entropies defined as

$\tilde{D}_{\alpha}(\rho,\sigma):=\frac{1}{2}(D_{\alpha}(\rho||\sigma)+D_{\alpha}(\sigma||\rho)),$
(3)

such that $\tilde{D}_{\alpha}(\rho,\sigma)=\tilde{D}_{\alpha}(\sigma,\rho)$,
where $D_{\alpha}(\rho||\sigma)$ is the Petz-Rényi relative entropy Petz
(1986) defined as

$D_{\alpha}(\rho||\sigma):=\frac{1}{\alpha-1}\ln\text{tr}(\rho^{\alpha}\sigma^{1-\alpha}),$
(4)

which is non negative for any $\alpha\in(0,1)\cup(1,\infty)$. It has
applications in quantum hypothesis testing Buscemi and Datta (2010); Brandao
and Datta (2011) and in quantum field theory Kudler-Flam (2023). The case
$\alpha=1$ is understood as
$D_{1}(\rho,\sigma):=(D(\rho||\sigma)+D(\sigma||\rho))/2$, where
$D(\rho||\sigma)=\text{tr}(\rho\ln\rho)-\text{tr}(\rho\ln\sigma)$ is the
quantum relative entropy. For the specific case $\alpha=1/2$, one has
$\tilde{D}_{1/2}(\rho,\sigma)=D_{1/2}(\rho||\sigma)=-2\ln\text{tr}(\sqrt{\rho}\sqrt{\sigma})=-\ln
F_{H}(\rho,\sigma)$ from (1) and (3). In this case, one could write Holevo’s
inequality (2) as a lower bound for $\tilde{D}_{1/2}(\rho,\sigma)$ as

$\tilde{D}_{1/2}(\rho,\sigma)\geq\ln\frac{1}{1-T(\rho,\sigma)^{2}},$ (5)

also recently used in the applications of Petz-Rényi relative entropy in
quantum field theory Kudler-Flam (2023), where it was explored a
Pinsker’s-like inequality,

$\tilde{D}_{\alpha}(\rho,\sigma)\geq 2\min(\alpha,1)T(\rho,\sigma)^{2}.$ (6)

Note that, for the case $\alpha=1/2$, Holevo’s inequality improves Pinsker’s
inequality for the particular case $\alpha=1/2$,

$\tilde{D}_{1/2}(\rho,\sigma)\geq\ln\frac{1}{1-T(\rho,\sigma)^{2}}\geq
T(\rho,\sigma)^{2}.$ (7)

Motivated by (7), we analyze the following question: can we generalize
Holevo’s inequality (5) beyond the case $\alpha=1/2$ such that it improves
Pinsker’s inequality for any $\alpha$ in the symmetric Petz-Rényi family? In
this case, we obtain the following result:

(Generalized Holevo’s inequality) Let $\rho$ and $\sigma$ be any density
matrices. Then,

$\tilde{D}_{\alpha}(\rho,\sigma)\geq\frac{1}{\alpha-1}\ln\frac{\cosh[(2\alpha-1)\operatorname{atanh}(T(\rho,\sigma))]}{\cosh[\operatorname{atanh}(T(\rho,\sigma))]}.$
(8)

For the specific case $\alpha=1/2$, our result (8) reproduces Holevo’s
inequality (5). Notably, the case $\alpha=1$ previously appeared in the
classic case ($[\rho,\sigma]=0$) as a bound for the symmetric Kullback-Leibler
divergence in terms of total variation Dechant (2022); Vo  _et al._ (2022);
Salazar (2022a). We also show that (8) improves Pinsker’s inequality (6) for
any $\alpha$.

More generally, we show that (8) is a consequence of a quantum uncertainty
relation for the symmetric Petz-Rényi relative entropy. This uncertainty
relation not only results in (8), but it also results in multiple relations
from stochastic and quantum thermodynamics as discussed in this letter. We
state our main result:

_Theorem:_ (Symmetric Petz-Rényi uncertainty relation) Let $\rho$ and $\sigma$
be any density matrices and $\hat{\theta}$ be any Hermitian operator. Then,

$\tilde{D}_{\alpha}(\rho,\sigma)\geq\frac{1}{\alpha-1}\ln\frac{\cosh[(2\alpha-1)\operatorname{atanh}(s(\rho,\sigma;\hat{\theta}))]}{\cosh[\operatorname{atanh}(s(\rho,\sigma;\hat{\theta}))]},$
(9)

where

$s(\rho,\sigma;\hat{\theta}):=\Big{[}\frac{(1/2)(\langle\hat{\theta}\rangle_{\rho}-\langle\hat{\theta}\rangle_{\sigma})^{2}}{\langle\langle\hat{\theta}\rangle\rangle_{\rho}+\langle\langle\hat{\theta}\rangle\rangle_{\sigma}+(1/2)(\langle\hat{\theta}\rangle_{\rho}-\langle\hat{\theta}\rangle_{\sigma})^{2}}\Big{]}^{1/2},$
(10)

where $\langle\hat{\theta}\rangle_{x}:=\text{tr}\\{x\hat{\theta}\\}$ and
$\langle\langle\hat{\theta}\rangle\rangle_{x}:=\text{tr}\\{x\hat{\theta}^{2}\\}-\text{tr}\\{x\hat{\theta}\\}^{2}$.
Note that the rhs in (9) depends only on the statistics of $\hat{\theta}$ with
respect to $\rho$ and $\sigma$ encoded in $s(\rho,\sigma;\hat{\theta})$, which
is the main idea behind uncertainty relations discussed below. As a
consequence of (9), we obtain (8) and other results similar to thermodynamic
uncertainty relations (TURs) Barato and Seifert (2015); Gingrich  _et al._
(2016); Polettini  _et al._ (2016); Pietzonka and Seifert (2017); Hasegawa and
Van Vu (2019a, b); Vo  _et al._ (2020); Vu and Hasegawa (2020); Timpanaro  _et
al._ (2019); Liu  _et al._ (2020); Horowitz and Gingrich (2020); Potts and
Samuelsson (2019); Proesmans and Horowitz (2019); Francica (2022); Salazar
(2022b); Brandner  _et al._ (2018); Carollo  _et al._ (2019); Liu and Segal
(2019); Van Vu and Saito (2022); Miller  _et al._ (2021); Pires  _et al._
(2021); Guarnieri  _et al._ (2019); Hasegawa (2020, 2021a, 2023, 2021b).

This letter is organized as follows. First, we present the main steps of the
proof, then we show how the bound is saturated and the improvement over
Pinsker’s inequality. Finally, we discuss the applications of (9) and obtain
multiple previous and new results in stochastic and quantum thermodynamics.

_Formalism:_ The idea behind the proof is a strategy that maps $n$ dimensional
quantum states $\rho=\sum_{i}p_{i}|p_{i}\rangle\langle p_{i}|$ and
$\sigma=\sum_{j}q_{j}|q_{j}\rangle\langle q_{j}|$ into the following $n^{2}$
dimensional classic distributions: $P_{ij}:=p_{i}|\langle
p_{i}|q_{j}\rangle|^{2}$ and $Q_{ij}:=q_{j}|\langle p_{i}|q_{j}\rangle|^{2}$.
This strategy is sometimes called Nussbaum-Szkoła distributions Nussbaum and
Szkoła (2009) and it was successfully used in other contexts Datta  _et al._
(2013); Salazar (2024), particularly as tool to calculate relative entropies
Androulakis and John (2023a, b) as follows,

$\sum_{ij}P_{ij}^{\alpha}Q_{ij}^{1-\alpha}=\sum_{ij}p_{i}^{\alpha}q_{j}^{1-\alpha}|\langle
p_{i}|q_{j}\rangle|^{2}=\text{tr}(\rho^{\alpha}\sigma^{1-\alpha}),$ (11)

which results in $D_{\alpha}(P|Q)=D_{\alpha}(\rho||\sigma)$, where
$D_{\alpha}(P|Q):=[1/(1-\alpha)]\ln\sum_{ij}P_{ij}^{\alpha}Q_{ij}^{1-\alpha}$
is the Rényi relative entropy, thus we have

$\tilde{D}_{\alpha}(P,Q)=\tilde{D}_{\alpha}(\rho,\sigma),$ (12)

where $\tilde{D}_{\alpha}(P,Q)=(1/2)[D_{\alpha}(P|Q)+D_{\alpha}(Q|P)]$. Now
that we mapped the Petz-Rényi relative entropy into a classic divergence, we
import a recent result from information theory Nishiyama (2022),

$\tilde{D}_{\alpha}(P,Q)\geq B(\alpha,\sqrt{\delta(P,Q)}),$ (13)

where $\delta(P,Q):=\sum_{s}(P_{s}-Q_{s})^{2}/(P_{s}+Q_{s})$ is the triangular
discrimination and

$B(\alpha,x):=\frac{1}{\alpha-1}\ln\frac{\cosh[(2\alpha-1)\operatorname{atanh}(x)]}{\cosh[\operatorname{atanh}(x)]}$
(14)

is increasing in the interval $0<x<1$ for all $\alpha>0$. Finally, we use the
inequality $\sqrt{\delta(P,Q)}\geq s(\rho,\sigma,\hat{\theta})$ (see
Appendix), which results in

$\tilde{D}_{\alpha}(\rho,\sigma)=\tilde{D}_{\alpha}(P,Q)\geq
B(\alpha,\sqrt{\delta(P,Q)})\geq B(\alpha,s(\rho,\sigma;\hat{\theta})),$ (15)

which proves our main result (9). The generalized Holevo’s inequality (8) is
obtained as a particular case of (9) where
$\hat{\theta}=\sum_{k}\operatorname{sign}(w_{k})|w_{k}\rangle\langle w_{k}|$
and $\\{|w_{k}\rangle\\}$ are the eigenvalues of $\rho-\sigma$ (see Appendix).

As in previous results, the bound (9) is saturated for the specific two-level
system, where $\rho=[e^{\epsilon/2}|1\rangle\langle
1|+e^{-\epsilon/2}|0\rangle\langle 0|]/(2\cosh(\epsilon/2))$,
$\sigma=[e^{-\epsilon/2}|1\rangle\langle 1|+e^{\epsilon/2}|0\rangle\langle
0|]/(2\cosh(\epsilon/2))$ and $\hat{\theta}=\phi(|1\rangle\langle
1|-|0\rangle\langle 0|)$. In this case, one has
$\text{tr}(\rho\hat{\theta})=\phi\tanh(\epsilon/2)$,
$\text{tr}(\sigma\hat{\theta})=-\phi\tanh(\epsilon/2)$ and
$\text{tr}(\rho\hat{\theta}^{2})=\text{tr}(\sigma\hat{\theta}^{2})=\phi^{2}$,
such that

$\tilde{D}_{\alpha}(\rho,\sigma)=\frac{1}{1-\alpha}\ln\frac{\cosh[(2\alpha-1)\epsilon/2]}{\cosh[\epsilon/2]},$
(16)

and also $s(\rho,\sigma;\hat{\theta})=|\tanh(\epsilon/2)|$. Therefore, in this
minimal system, we have the saturation of (9),

$\tilde{D}_{\alpha}(\rho,\sigma)=B(\alpha,s(\rho,\sigma;\hat{\theta})).$ (17)

We also show that (8) improves Pinsker’s inequality (6) in the Appendix, where
we have

$\tilde{D}_{\alpha}(\rho,\sigma)\geq B(\alpha,T(\rho,\sigma))\geq
2\min(\alpha,1)T(\rho,\sigma)^{2},$ (18)

which generalizes Holevo’s case (7) for any $\alpha$.

_Discussion -_ We investigate particular cases of (9) and (8) in stochastic
and quantum thermodynamics. Some of the cases are very well known and some of
them are new to our knowledge.

First, turning our attention to relation (8), we note that particular cases
also appeared in the literature. The case $\alpha=1/2$ is obviously the
Holevo’s inequality (5), but we also observe that the case
$\lim\alpha\rightarrow 1$ results in

$\tilde{D}(\rho,\sigma)\geq
2T(\rho,\sigma)\operatorname{atanh}(T(\rho,\sigma)),$ (19)

which was recently used to to analyse fluxes in quantum thermodynamics Salazar
(2023a). We also note that the classic case ($[\rho,\sigma]=0$) of (19) was
used in the study of Markov chains in stochastic thermodynamics as well
Dechant (2022); Vo  _et al._ (2022); Salazar (2022a), where $\rho$ and
$\sigma$ can be written as classic probabilities $p=(p_{1},...,p_{n})$,
$q=(q_{1},...,q_{n})$, yielding

$\tilde{D}(p,q)\geq 2\Delta(p,q)\operatorname{atanh}(\Delta(p,q)),$ (20)

where $\Delta(p,q)=(1/2)\sum_{s}|p_{i}-q_{i}|$ is the total variation.

Moreover, we note that (9) can be inverted to the following expression when
$\langle\hat{\theta}\rangle_{\rho}\neq\langle\hat{\theta}\rangle_{\sigma}$,

$\frac{\langle\langle\hat{\theta}\rangle\rangle_{\rho}+\langle\langle\hat{\theta}\rangle\rangle_{\sigma}}{(1/2)(\langle\hat{\theta}\rangle_{\rho}-\langle\hat{\theta}\rangle_{\sigma})^{2}}\geq
f(\alpha,\tilde{D}_{\alpha}(\rho,\sigma)),$ (21)

where $f(\alpha,x):=1/[B^{-1}(\alpha,D)]^{2}-1$, and for a fixed $\alpha$,
$B^{-1}(\alpha,x)$ is the inverse of $B(\alpha,x)$ for $x\geq 0$, such that
$B^{-1}(\alpha,B(\alpha,x))=x$. Remarkably, expression (21) has the form of a
quantum uncertainty relation, which explains the name of the theorem as
symmetric Petz-Rényi uncertainty relation. As matter of fact, the case
$\alpha=1$ results in the recently proposed quantum relative entropy
uncertainty relation Salazar (2024),

$\frac{\langle\langle\hat{\theta}\rangle\rangle_{\rho}+\langle\langle\hat{\theta}\rangle\rangle_{\sigma}}{(1/2)(\langle\hat{\theta}\rangle_{\rho}-\langle\hat{\theta}\rangle_{\sigma})^{2}}\geq
f(1,\tilde{D}(\rho,\sigma)),$ (22)

where $f(1,x)=1/\sinh^{2}(g(x)/2)$ and $g(x)$ is the inverse of
$h(x)=x\tanh(x/2)$ for $x\geq 0$. Analogously, the case $\alpha=1/2$ in (21)
results in the following Holevo’s uncertainty relation,

$\frac{\langle\langle\hat{\theta}\rangle\rangle_{\rho}+\langle\langle\hat{\theta}\rangle\rangle_{\sigma}}{(1/2)(\langle\hat{\theta}\rangle_{\rho}-\langle\hat{\theta}\rangle_{\sigma})^{2}}\geq\frac{F_{H}(\rho,\sigma)}{1-F_{H}(\rho,\sigma)},$
(23)

which takes the usual form of thermodynamic uncertainty relations as a lower
bound for some uncertainty in terms of a dissimilarity. In this case, the
dissimilarity is not the usual entropy production, but given in terms of the
Holevo’s fidelity (1) instead.

We also consider the classic situation $[\rho,\sigma]=0$, which represents the
absence of coherence between the states. In this particular case, writing
again $\rho$ and $\sigma$ in terms of classic probabilities
$p=(p_{1},...,p_{n})$, $q=(q_{1},...,q_{n})$ and
$\theta=(\theta_{1},...,\theta_{n})$ is a random variable. We obtain the
classic version of (21),

$\frac{\langle\langle\theta\rangle\rangle_{p}+\langle\langle\theta\rangle\rangle_{q}}{(1/2)(\langle\theta\rangle_{p}-\langle\theta\rangle_{q})^{2}}\geq
f(\alpha,\tilde{D}_{\alpha}(p,q)),$ (24)

which can be seen as the $\alpha-$generalized version of the tightest form of
the hysteretic thermodynamic uncertainty relation ($\alpha=1$) Proesmans and
Horowitz (2019); Francica (2022); Salazar (2022b),

$\frac{\langle\langle\theta\rangle\rangle_{p}+\langle\langle\theta\rangle\rangle_{q}}{(1/2)(\langle\theta\rangle_{p}-\langle\theta\rangle_{q})^{2}}\geq
f(1,\tilde{D}(p,q)),$ (25)

while the case $\alpha=1/2$ in (24) can be written as (23) but in terms of the
Bhattacharyya or Hellinger distances.

Finally, in the particular case $p=P(\Gamma)$, $q=P(\Gamma^{\dagger})$, where
$\Gamma$ typically represents a trajectory and $\Gamma^{\dagger}$ is the
inverse trajectory, such that $(\Gamma^{\dagger})^{\dagger}=\Gamma$, and
$\theta(\Gamma)$ is a current with property
$\theta(\Gamma^{\dagger})=-\theta(\Gamma)$, relation (25) yields a result
known as the thermodynamic uncertainty relation from the exchange fluctuation
theorem Hasegawa and Van Vu (2019a); Timpanaro  _et al._ (2019),

$\frac{\langle\langle\theta\rangle\rangle_{p}}{\langle\theta\rangle_{p}^{2}}\geq
f(1,\langle\Sigma\rangle),$ (26)

where $\langle\Sigma\rangle:=\sum_{\Gamma}P(\Gamma)\ln
P(\Gamma)/P(\Gamma^{\dagger})$ is the average entropy production.

_Conclusions -_ We proposed a symmetric Petz-Rényi uncertainty relation (9)
and studied multiple applications. Our result was obtained using a mapping
from quantum to classic systems and exploring recent results from information
theory. From our result, we wrote a general uncertainty relation (21) and, for
the particular case $\alpha=1/2$, we obtained a uncertainty relation in terms
of Holevo’s fidelity (22), a generalized Holevo’s inequality (8) and multiple
known results in quantum and stochastic thermodynamics (20,25,26). Our results
highlight that different symmetric quantum dissimilarities may also play a
role akin to the entropy production in expressions resembling thermodynamic
uncertainty relations. This fact does not depend on specific properties of the
system (such as the detailed fluctuation theorem), but it is rather a
fundamental interplay between quantum uncertainties and symmetric
dissimilarities.

_Appendix-_ We start proving a result in information theory for probabilities
($P,Q$). Then, we will show the quantum case is obtained as a consequence of
the classic case using a strategy that maps $n$dimensional quantum states into
$n^{2}$ dimensional classic distributions.

_Definition 1._ Let $P=\\{P_{s}\\}$ and $Q=\\{Q_{s}\\}$ be probabilities in a
set $S$, and let be the triangular discrimination be defined as

$\delta(P,Q):=\sum_{s}\frac{(P_{s}-Q_{s})^{2}}{(P_{s}+Q_{s})},$ (27)

with the notation $0^{2}/0=0$ for the cases where $P_{s}=Q_{s}$. We define the
Rényi relative entropy as

$D_{\alpha}(P|Q):=\frac{1}{\alpha-1}\ln\sum_{s}P_{s}^{\alpha}Q_{s}^{1-\alpha},$
(28)

for $\alpha\in(0,1)\cup(1,\infty)$ and the symmetric version,

$\tilde{D}_{\alpha}(P,Q):=\frac{1}{2}(D_{\alpha}(P|Q)+D_{\alpha}(Q|P)).$ (29)

_Lemma 1._ Let $P$ and $Q$ be any distributions, then

$\tilde{D}_{\alpha}(P,Q)\geq\frac{1}{\alpha-1}\ln\frac{\cosh[(2\alpha-1)\operatorname{atanh}(\sqrt{\delta(P,Q)})]}{\cosh[\operatorname{atanh}(\sqrt{\delta(P,Q)}]}.$
(30)

Proof. This is a particular case of a recent result for $f-$divergences
Nishiyama (2022). We provide an alternative poof using a formulation in terms
of stochastic entropy Salazar (2022b, 2023b). Let $\\{P_{s}\\}$ be a
probability function and let $s^{\prime}:=m(s)$ be any involution (such that
$m(m(s))=s$) and define a new probability $P^{\prime}(s):=P(s^{\prime})$.
Unless mentioned otherwise, expectations $\langle\rangle$ are meant with
respect to $P_{s}$, $\langle\rangle=\langle\rangle_{P}$. If the pair
($P,P^{\prime}$) is not absolutely continuous (ie, there is an $s$ such that
$P_{s}^{\prime}=0$ and $P_{s}>0$), then $\tilde{D}_{\alpha}(P,Q)=\infty$ and
(30) holds immediately. Thus, we focus on the cases where $(P,P^{\prime})$ are
absolutely continuous ($P_{s}=0\leftrightarrow P_{s}^{\prime}=0$, for all
$s\in S$). Let $\Sigma(s)$ be an entropy-like random variable defined as

$\Sigma(s):=\ln\frac{P(s)}{P^{\prime}(s)},$ (31)

when $P_{s}^{\prime}\neq 0$ and $\Sigma(s)=0$ otherwise. In the stochastic
thermodynamics literature, this is known as the strong detailed fluctuation
theorem and $\Sigma$ is the stochastic entropy production, although in our
notation (31) is just a definition of a random variable $\Sigma(s)$. Consider
the following expectation with respect to $P$,

$\langle\exp(\beta\Sigma)\rangle=\sum_{s}P(s)^{\beta+1}P^{\prime}(s)^{-\beta},$
(32)

which can also be rewritten as
$\exp(\beta\Sigma)=\sinh(\beta\Sigma)+\cosh(\beta\Sigma)$. Now using the
following property, $\langle u(\Sigma)\rangle=\langle
u(\Sigma)\tanh(\Sigma/2)\rangle$ for odd functions $u(-x)=-u(x)$, we obtain

$\langle\exp(\beta\Sigma)\rangle=\langle\sinh(\beta\Sigma)\tanh(\Sigma/2)+\cosh(\beta\Sigma)\rangle,$
(33)

which has the following compact form from
$\cosh(x+y)=\cosh(x)\cosh(y)+\sinh(x)\sinh(y)$ and replacing $\beta=\alpha-1$,

$\langle\exp((\alpha-1)\Sigma)\rangle=\langle\frac{\cosh[(\alpha-1/2)\Sigma]}{\cosh(\Sigma/2)}\rangle.$
(34)

Now we note that the triangular discrimination $\delta(P,P^{\prime})$ has a
particular form in terms of the statistics of $\Sigma(s)$ as

$\delta(P,P^{\prime})=\sum_{s}\frac{(P_{s}-P_{s}^{\prime})^{2}}{(P_{s}+P_{s}^{\prime})^{2}}\frac{P_{s}+P_{s}^{\prime}}{2}=\langle\tanh(\Sigma/2)^{2}\rangle,$
(35)

using the fact that
$\langle\tanh(\Sigma/2)^{2}\rangle_{(P+P^{\prime})/2}=\langle\tanh(\Sigma/2)^{2}\rangle_{P}$.
Finally, we write the function
$F(\Sigma):=\cosh[(\alpha-1/2)\Sigma]/\cosh(\Sigma/2)$ from (35) in terms of
$\tanh(x/2)^{2}$

$F(\Sigma)=F(2\arctan(\sqrt{\tanh(\Sigma/2)^{2}})),$ (36)

and use Jensen’s inequality in (36)

$\displaystyle\langle F(\Sigma)\rangle\geq
F(2\arctan(\sqrt{\langle\tanh(\Sigma/2)^{2}\rangle}))=$
$\displaystyle=F(2\operatorname{atanh}(\sqrt{\delta(P,P^{\prime})}),$ (37)

since $d^{2}F(2\operatorname{atanh}(\sqrt{y}))/dy^{2}\geq 0$ for $\alpha>1$.
Similarly, we get $\langle F(\Sigma)\rangle\leq
F(2\arctan(\sqrt{\langle\tanh(\Sigma/2)^{2}\rangle}))$ for $0\leq\alpha<1$.
Combining (34) and (Symmetric Petz-Rényi relative entropy uncertainty
relation), we obtain

$\langle\exp(\alpha-1)\Sigma\rangle\geq\frac{\cosh[(2\alpha-1)\operatorname{atanh}(\sqrt{\delta(P,P^{\prime})}]}{\cosh(\operatorname{atanh}(\sqrt{\delta(P,P^{\prime})})},$
(38)

for $\alpha>1$ and $\langle\exp(\alpha-1)\Sigma\rangle\leq
F(2\operatorname{atanh}(\sqrt{\delta(P,Q)}))$ for $\alpha<1$. Note that Rényi
relative entropy is given by

$D_{\alpha}(P|P^{\prime})=\frac{1}{\alpha-1}\ln\langle\exp(\alpha-1)\Sigma\rangle,$
(39)

so that the cases $\alpha>1$ in (39) and the case $\alpha<1$ can be rewritten
as a single expression,

$D_{\alpha}(P|P^{\prime})\geq\frac{1}{\alpha-1}\ln\frac{\cosh[(2\alpha-1)\operatorname{atanh}(\sqrt{\delta(P,P^{\prime})})]}{\cosh(\operatorname{atanh}(\sqrt{\delta(P,P^{\prime})})},$
(40)

for all $\alpha\neq 1$. The case $\alpha=1$ should be understood as the limit
$\alpha\rightarrow 1$. Since (40) was proved for any probability $P$ and any
involution $m(s)$, we apply the expression for the particular set
$\\{(s,i)\\}$, where $s\in S$ and $i\in\\{0,1\\}$, with involution
$m(s,i)=(s,m(i))$, where $m(i)=1-i$ and probabilities $p(s,1):=P(s)/2$ and
$p(s,0)=Q(s)/2$. In this case, we obtain
$D_{\alpha}(p|p^{\prime})=(1/2)(D_{\alpha}(P|Q)+D_{\alpha}(Q|P))=\tilde{D}_{\alpha}(P,Q)$
and $\delta(p,p^{\prime})=\delta(P,Q)$, which results in our Lemma 1 (30) for
any $P,Q$. $\blacksquare$

_Lemma 2._ Let $\Theta_{s}\in\mathbb{C}$ be a complex random variable, then

$\delta(P,Q)\geq\frac{(1/2)|\langle\Theta\rangle_{P}-\langle\Theta\rangle_{Q}|^{2}}{\langle\langle\Theta\rangle\rangle_{P}+\langle\langle\Theta\rangle\rangle_{Q}+(1/2)|\langle\Theta\rangle_{P}-\langle\Theta\rangle_{Q}|^{2}},$
(41)

where
$\langle\langle\Theta\rangle\rangle:=\langle|\Theta|^{2}\rangle-|\langle\Theta\rangle|^{2}$.

Proof. This a is a complex generalization of a previous result Falasco  _et
al._ (2022) also used in Salazar (2023a) and we reproduce here. Consider
probabilities $P,Q$ in $s\in S$, $\sum_{s}P(s)=\sum_{s}Q(s)=1$ and a complex
valued random variable $\Theta(s)\in\mathbb{C}$. We define $S^{\prime}=\\{s\in
S|P(s)+Q(s)>0\\}$ and the probability $\tilde{P}(s):=(P(s)+Q(s))/2$ in
$S^{\prime}$, $\sum_{s\in S^{\prime}}\tilde{P}(s)=1$,
$\overline{\Theta}_{X}:=\langle\Theta\rangle_{X}=\sum_{s}\Theta(s)X(s)$, for
$X\in\\{P,Q,\tilde{P}\\}$. Note that the expression
$|\overline{\Theta}_{P}-\overline{\Theta}_{Q}|^{2}$ can be rewritten as

$\frac{1}{4}|\overline{\Theta}_{P}-\overline{\Theta}_{Q}|^{2}=|\sum_{s\in
S^{\prime}}(\Theta(s)-c)\frac{(P(s)-Q(s))}{2}|^{2},$ (42)

for any complex $c$. Using Cauchy-Schwarz inequality, we also obtain for any
complex $c$,

$|\sum_{s\in
S^{\prime}}(\Theta(s)-c)\frac{(P(s)-Q(s))}{2}|^{2}\leq\langle|\Theta-c|^{2}\rangle_{\tilde{P}}\langle(\frac{P-Q}{P+Q})^{2}\rangle_{\tilde{P}},$
(43)

so that combining (42) and (43) for $c=\overline{\Theta}_{\tilde{P}}$, it
yields

$\frac{1}{4}|\overline{\Theta}_{P}-\overline{\Theta}_{Q}|^{2}\leq\langle|\Theta-\overline{\Theta}_{\tilde{P}}|^{2}\rangle_{\tilde{P}}\langle(\frac{P-Q}{P+Q})^{2}\rangle_{\tilde{P}}.$
(44)

We note from Definition 1 (27) that

$\delta(P,Q)=\langle(\frac{P-Q}{P+Q})^{2}\rangle_{\tilde{P}}.$ (45)

Now consider the identity

$4\langle|\Theta-\overline{\Theta}_{\tilde{P}}|^{2}\rangle_{\tilde{P}}=2(\langle|\Theta|^{2}\rangle_{P}-|\overline{\Theta}_{P}|^{2})+2(\langle|\Theta|^{2}\rangle_{Q}-|\overline{\Theta}_{Q}|^{2})+|\overline{\Theta}_{P}-\overline{\Theta}_{Q}|^{2}.$
(46)

Combining (44), (45) and (46) it results in (41) $\blacksquare$.

_Definition 2._ (Nussbaum-Szkoła distributions) Let $\rho$ and $\sigma$ be
density matrices in $n$ dimensions with spectral decomposition
$\rho=\sum_{i}^{n}p_{i}|p_{i}\rangle\langle p_{i}|$ and
$\sigma=\sum_{j}^{n}q_{j}|q_{j}\rangle\langle q_{j}|$. We define distributions
$P$ and $Q$ in $S=\\{(i,j)|1\leq i\leq n;1\leq j\leq n\\}$ as

$P_{ij}:=|\langle p_{i}|q_{j}\rangle|^{2}p_{i},$ (47) $Q_{ij}:=|\langle
p_{i}|q_{j}\rangle|^{2}q_{j},$ (48)

and we add the definition of the auxiliary complex random variable $\Theta$
based on a Hermitian operator $\hat{\theta}$,

$\Theta_{ij}:=\frac{\langle p_{i}|\hat{\theta}|q_{j}\rangle}{\langle
p_{i}|q_{j}\rangle},$ (49)

for $\langle p_{i}|q_{j}\rangle\neq 0$ and $\Theta_{ij}:=0$ otherwise.

_Lemma 3._ For any density matrices $\rho,\sigma$ and Hermitian operator
$\hat{\theta}$ we have

$\displaystyle\frac{(1/2)|\langle\Theta\rangle_{P}-\langle\Theta\rangle_{Q}|^{2}}{\langle\langle\Theta\rangle\rangle_{P}+\langle\langle\Theta\rangle\rangle_{Q}+(1/2)|\langle\Theta\rangle_{P}-\langle\Theta\rangle_{Q}|^{2}}$
$\displaystyle\geq\frac{(1/2)(\langle\hat{\theta}\rangle_{\rho}-\langle\hat{\theta}\rangle_{\sigma})^{2}}{\langle\langle\hat{\theta}\rangle\rangle_{\rho}+\langle\langle\hat{\theta}\rangle\rangle_{\sigma}+(1/2)(\langle\hat{\theta}\rangle_{\rho}-\langle\hat{\theta}\rangle_{\sigma})^{2}},$
(50)

with $P,Q,\Theta$ given by Definition 2 and
$\langle\hat{\theta}\rangle_{\rho}:=\text{tr}\\{\rho\hat{\theta}\\}$,
$\langle\hat{\theta}\rangle_{\sigma}:=\text{tr}\\{\sigma\hat{\theta}\\}$,
$\langle\langle\hat{\theta}\rangle\rangle_{\rho}:=\text{tr}\\{\rho\hat{\theta}^{2}\\}-\text{tr}\\{\rho\hat{\theta}\\}^{2}$
and
$\langle\langle\hat{\theta}\rangle\rangle_{\sigma}:=\text{tr}\\{\sigma\hat{\theta}^{2}\\}-\text{tr}\\{\sigma\hat{\theta}\\}^{2}$.

Proof. This idea was explored in Salazar (2024). The expected value of
$\hat{\theta}$ with respect to $\rho$ is

$\text{tr}(\rho\hat{\theta})=\sum_{ij}p_{i}\langle
p_{i}|\hat{\theta}|q_{j}\rangle\langle q_{j}|p_{i}\rangle=\sum_{ij;\langle
q_{j}|p_{i}\rangle\neq 0}p_{i}|\langle q_{j}|p_{i}\rangle|^{2}\frac{\langle
p_{i}|\hat{\theta}|q_{j}\rangle}{\langle p_{i}|q_{j}\rangle},$ (51)

where we used $\langle p_{i}|q_{j}\rangle=\langle q_{j}|p_{i}\rangle^{*}$.
Using Definition 2, in terms of $P$ and $\Theta$, we have from (51),

$\text{tr}(\rho\hat{\theta})=\sum_{ij}P_{ij}\Theta_{ij}:=\langle\Theta\rangle_{P}.$
(52)

Similarly, we obtain for the expected value of $\hat{\theta}$ with respect to
$\sigma$ using Definition 2,

$\text{tr}(\sigma\hat{\theta})=\sum_{ij}Q_{ij}\Theta_{ij}:=\langle\Theta\rangle_{Q}.$
(53)

Analogously, we have for the expected value of $\hat{\theta}^{2}$ with respect
to $\rho$,

$\displaystyle\text{tr}(\rho\hat{\theta}^{2})=\sum_{ij}p_{i}|\langle
p_{i}|\hat{\theta}|q_{j}\rangle|^{2}\geq$ (54) $\displaystyle\sum_{ij;\langle
q_{j}|p_{i}\rangle\neq 0}p_{i}|\langle
p_{i}|\hat{\theta}|q_{j}\rangle|^{2}=\sum_{ij}P_{ij}|\Theta_{ij}|^{2},$ (55)

where we used $\hat{\theta}=\hat{\theta}^{\dagger}$, which yields

$\text{tr}(\rho\hat{\theta}^{2})\geq\sum_{ij}P_{ij}|\Theta_{ij}|^{2}:=\langle|\Theta|^{2}\rangle_{P}.$
(56)

We have a similar expression in terms of $\sigma$,

$\text{tr}(\sigma\hat{\theta}^{2})\geq\sum_{ij}Q_{ij}|\Theta_{ij}|^{2}:=\langle|\Theta|^{2}\rangle_{Q}.$
(57)

Combining expressions (52), (53), (56) and (57) completes the proof of Lemma 3
(Symmetric Petz-Rényi relative entropy uncertainty relation). $\blacksquare$

_Definition 3._ Let the trace-norm be defined as

$T(\rho,\sigma):=\frac{1}{2}|\rho-\sigma|_{1}=\frac{1}{2}\text{tr}\\{\sqrt{(\rho-\sigma)^{2}}\\},$
(58)

and the Petz-Rényi relative entropy

$D_{\alpha}(\rho||\sigma):=\frac{1}{\alpha-1}\ln\text{tr}\\{\rho^{\alpha}\sigma^{1-\alpha}\\},$
(59)

for any $\alpha\in(0,1)\cup(1,\infty)$, with the symmetric version defined as

$\tilde{D}_{\alpha}:=\frac{1}{2}(D_{\alpha}(\rho||\sigma)+D_{\alpha}(\sigma||\rho)).$
(60)

_Theorem 1._ (Symmetric Petz-Rényi uncertainty relation) Let $\rho$ and
$\sigma$ be any density matrices and $\hat{\theta}$ be any Hermitian operator.
Then,

$\tilde{D}_{\alpha}(\rho,\sigma)\geq\frac{1}{\alpha-1}\ln\frac{\cosh[(2\alpha-1)\operatorname{atanh}(s(\rho,\sigma;\hat{\theta}))]}{\cosh[\operatorname{atanh}(s(\rho,\sigma;\hat{\theta}))]},$
(61)

where

$s(\rho,\sigma;\hat{\theta}):=\Big{[}\frac{(1/2)(\langle\hat{\theta}\rangle_{\rho}-\langle\hat{\theta}\rangle_{\sigma})^{2}}{\langle\langle\hat{\theta}\rangle\rangle_{\rho}+\langle\langle\hat{\theta}\rangle\rangle_{\sigma}+(1/2)(\langle\hat{\theta}\rangle_{\rho}-\langle\hat{\theta}\rangle_{\sigma})^{2}}\Big{]}^{1/2}.$
(62)

Proof. Using Definition 2, we have the following identity

$\sum_{ij}P_{ij}^{\alpha}Q_{ij}^{1-\alpha}=\sum_{ij}p_{i}^{\alpha}q_{j}^{1-\alpha}|\langle
p_{i}|q_{j}\rangle|^{2}=\text{tr}(\rho^{\alpha}\sigma^{1-\alpha}),$ (63)

which results in $D_{\alpha}(P|Q)=D_{\alpha}(\rho||\sigma)$, thus

$\tilde{D}_{\alpha}(P,Q)=\tilde{D}_{\alpha}(\rho,\sigma).$ (64)

We also have from Lemma 2 (41) and Lemma 3 (Symmetric Petz-Rényi relative
entropy uncertainty relation),

$\sqrt{\delta(P,Q)}\geq s(\rho,\sigma;\hat{\theta}).$ (65)

Finally, note that the function

$B(\alpha,x):=\frac{1}{\alpha-1}\ln\frac{\cosh[(2\alpha-1)\operatorname{atanh}(x)]}{\cosh[\operatorname{atanh}(x)]}$
(66)

is increasing for all $\alpha\in(0,1)\cup(1,\infty)$, which combined with
(64), (65) and Lemma 1 (30), it results in

$\tilde{D}_{\alpha}(\rho,\sigma)=\tilde{D}_{\alpha}(P,Q)\geq
B(\alpha,\sqrt{\delta(P,Q)})\geq B(\alpha,s(\rho,\sigma;\hat{\theta})),$ (67)

which proves Theorem 1 (61). $\blacksquare$

_Corollary 1._ (Generalized Holevo’s inequality) Let $\rho$ and $\sigma$ be
any density matrices and $\hat{\theta}$ be any Hermitian operator. Then,

$\displaystyle\tilde{D}_{\alpha}(\rho,\sigma)\geq\frac{1}{\alpha-1}\ln\frac{\cosh[(2\alpha-1)\operatorname{atanh}(T(\rho,\sigma))]}{\cosh[\operatorname{atanh}(T(\rho,\sigma))]}.$
(68)

Proof. We use Theorem 1, we use a specific operator
$\hat{\theta}=\hat{\omega}$ suggested in Salazar (2023a) as follows, consider
the following spectral decomposition of the operator
$\rho-\sigma=\sum_{k}w_{k}|w_{k}\rangle\langle w_{k}|$. Now we define
$\hat{\omega}=\hat{\omega}$ as

$\hat{\omega}:=\sum_{k,w_{k}\neq
0}\operatorname{sign}(w_{k})|w_{k}\rangle\langle w_{k}|,$ (69)

where $\operatorname{sign}(x)=1\leavevmode\nobreak\ (-1)$, for
$x>0\leavevmode\nobreak\ (x<0)$. Then, we write the trace-norm
$|\rho-\sigma|_{1}$ in terms of $\hat{\omega}$ from (69),

$\langle\hat{\omega}\rangle_{\rho}-\langle\hat{\omega}\rangle_{\sigma}=\text{tr}[\hat{\omega}(\rho-\sigma)]=\sum_{k}|w_{k}|=|\rho-\sigma|_{1}.$
(70)

Then, we observe that

$\hat{\omega}^{2}=\sum_{k,w_{k}\neq
0}\operatorname{sign}(w_{k})^{2}|w_{k}\rangle\langle w_{k}|=I-\hat{\epsilon},$
(71)

where $I$ is the identity operator and
$\hat{\epsilon}:=\sum_{k,w_{k}=0}|w_{k}\rangle\langle w_{k}|$ , with averages

$\langle\hat{\epsilon}\rangle_{\rho}=\langle\hat{\epsilon}\rangle_{\sigma}:=\epsilon,$
(72)

obtained from
$\langle\hat{\epsilon}\rangle_{\rho}-\langle\hat{\epsilon}\rangle_{\sigma}=\text{tr}[\hat{\epsilon}(\rho-\sigma)]=\sum_{k,w_{k}=0}w_{k}=0$.
We also have $0\leq\epsilon\leq 1$, because $\rho,\sigma$ are positive
definite and $\text{tr}(\rho)=\text{tr}(\sigma)=1$. From (71) and (72), we get

$\langle\hat{\omega}^{2}\rangle_{\rho}=\langle\hat{\omega}^{2}\rangle_{\sigma}=1-\epsilon.$
(73)

Using the averages (70) and (73) we obtain

$s(\rho,\sigma;\hat{\omega})^{2}=\frac{(1/2)|\rho-\sigma|_{1}^{2}}{(2-2\epsilon)-\langle\hat{\omega}\rangle_{\rho}-\langle\hat{\omega}\rangle_{\sigma}+(1/2)|\rho-\sigma|_{1}^{2}}.$
(74)

Also check that

$(1/2)|\rho-\sigma|_{1}^{2}\leq\langle\hat{\omega}\rangle_{\rho}^{2}+\langle\hat{\omega}\rangle_{\sigma}^{2},$
(75)

directly from (70) and the expression
$(1/2)(x-y)^{2}\leq(1/2)(x-y)^{2}+(1/2)(x+y)^{2}=x^{2}+y^{2}$, for
$x=\langle\hat{\omega}\rangle_{\rho}$,
$y=\langle\hat{\omega}\rangle_{\sigma}$. In this case, we obtain from (75),
$s(\rho,\sigma;\hat{\omega})^{2}\geq\frac{|\rho-\sigma|_{1}^{2}}{4(1-\epsilon)}\geq\frac{|\rho-\sigma|_{1}^{2}}{4}$,
which results in

$s(\rho,\sigma;\hat{\omega})\geq\frac{|\rho-\sigma|_{1}}{2}=T(\rho,\sigma).$
(76)

Now, using Theorem 1 with operator $\hat{\omega}$, we have
$\tilde{D}_{\alpha}(\rho,\sigma)\geq B(\alpha,s(\rho,\sigma;\hat{\omega}))\geq
B(\alpha,T(\rho,\alpha))$, using (76) and the fact that $B(\alpha,x)$ is
increasing, which proves the first inequality in (68). $\blacksquare$

_Lemma 4._ The bound in Corollary 1 improves Pinsker’s inequality.

Proof. Let $T\in[0,1]$, then we show that

$B(\alpha,T):=\frac{1}{\alpha-1}\ln\frac{\cosh[(2\alpha-1)\operatorname{atanh}(T)]}{\cosh[\operatorname{atanh}(T)]}\geq
2\min(\alpha,1)T^{2}.$ (77)

For that purpose, we consider the function
$g(\alpha,x)=\ln[\cosh((2\alpha-1)x)/\cosh(x)]$ for $x\geq 0$ and
$\alpha\in(0,1)\cup(1,\infty)$. For a constant $x$, we have
$\partial^{2}g(\alpha,x)/\partial^{2}\alpha=4x^{2}\operatorname{sech}^{2}[(x(1-2\alpha)]\geq
0$. Therefore, we could use the property of convex functions,
$g(\alpha,x)-g(1,x)\geq g^{\prime}(1,x)(\alpha-1)$, where
$g^{\prime}(\alpha,x)=\partial g(\alpha,x)/\partial\alpha$. Replacing
$g^{\prime}(1,x)=2x\tanh(x)$ and $g(1,x)=0$, we have $g(\alpha,x)\geq
2x\tanh(x)(\alpha-1)$, which yields for $\alpha>1$,

$B(\alpha,T)=\frac{1}{\alpha-1}g(\alpha,\operatorname{atanh}(T))\geq
2T\operatorname{atanh}(T)\geq 2T^{2}.$ (78)

Now for $0<\alpha<1$, we use a different strategy. First, note that
$g(\alpha,x)$ can be conveniently rewritten as
$g(\alpha,x)=2\operatorname{atanh}(\tanh(\alpha x)\tanh((\alpha-1)x))$ using
$\operatorname{atanh}(x)=(1/2)\ln[(1+x)/(1-x)]$. Thus we obtain using
$-\operatorname{atanh}(x)\leq-x$,

$g(\alpha,x)\leq-2\tanh(\alpha x)\tanh((1-\alpha)x)\leq
2\alpha(\alpha-1)\tanh(x)^{2},$ (79)

where we used $\tanh(\epsilon x)\geq\epsilon\tanh(x)$ for $0<\epsilon<1$ and
$x>0$ (one can check this showing the function $j(x)=\tanh(\epsilon
x)-\epsilon\tanh(x)$ is increasing in $x$ for any constant $\epsilon\in(0,1)$
and $j(0)=0$). Finally, we get from (79), using $\alpha-1<0$,

$B(\alpha,T)=\frac{1}{\alpha-1}g(\alpha,\operatorname{atanh}(T))\geq 2\alpha
T^{2}.$ (80)

Combining (78) and (80), it results in $B(\alpha,T)\geq 2\min(\alpha,1)T^{2}$
for all $\alpha\geq 0$, which proves (77). $\blacksquare$

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# Boolean Complexes of Involutions

Axel Hultman and Vincent Umutabazi  Department of Mathematics, Linköping
University, SE-581 83 Linköping, Sweden<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Let $(W,S)$ be a Coxeter system. We introduce the boolean complex of
involutions of $W$ which is an analogue of the boolean complex of $W$ studied
by Ragnarsson and Tenner. By applying discrete Morse theory, we determine the
homotopy type of the boolean complex of involutions for a large class of
$(W,S)$, including all finite Coxeter groups, finding that the homotopy type
is that of a wedge of spheres of dimension $|S|-1$. In addition, we find
simple recurrence formulas for the number of spheres in the wedge.

##  1\. Introduction

Let $F$ be a finite alphabet. A word over $F$ is injective if it is a sequence
of distinct letters from $F$. The poset of all injective words over $F$
ordered by the (not necessarily consecutive) subword relation is a simplicial
poset, hence the face poset of a boolean cell complex. This is the well-
studied complex of injective words. Farmer [2] showed that it is homotopy
equivalent to a wedge of spheres of dimension $|F|-1$. Jonsson and Welker [12]
proved the same conclusion for considerably more general classes of boolean
cell complexes using shellability techniques.

Ragnarsson and Tenner [15] independently arrived at one of Jonsson and
Welker’s families of complexes from a different starting point. Namely,
Ragnarsson and Tenner considered the boolean elements of a Coxeter system
$(W,S)$, i.e. those elements whose Bruhat order ideals are boolean lattices.
These elements were first considered by Tenner in [18]. Under Bruhat order,
they generate the face poset of a boolean cell complex, the boolean complex of
$(W,S)$. Using discrete Morse theory, it was shown in [15] that this complex
is homotopy equivalent to a wedge of spheres of dimension $|S|-1$, and a
recurrence formula for the number of spheres was found.

In [9], the first named author and Vorwerk studied the boolean involutions of
$(W,S)$ as an involution analogue of Tenner’s boolean elements. In the Bruhat
order on involutions, they too form the face poset of a boolean cell complex.
In this paper, we study this boolean complex of involutions,
$\Delta_{\mathrm{inv}}(W)$. Such complexes are not special cases of Jonsson
and Welker’s constructions.

For a large class of Coxeter systems, including all finite ones, we apply
discrete Morse theory in order to show that $\Delta_{\mathrm{inv}}(W)$ is also
homotopy equivalent to a wedge of spheres of dimension $|S|-1$ and provide a
recurrence formula for the number of spheres.

The remainder of this paper is organized in the following way. Section 2
contains preliminaries on cell complexes, acyclic matchings, and Coxeter group
theory. Section 3 contains the main technical result of this paper, Theorem
3.4. In Section 4, we provide applications of the main result by computing the
homotopy type of $\Delta_{\mathrm{inv}}(W)$ for many Coxeter groups, including
all finite types.

##  2\. Preliminaries

In this section, notions on simplicial posets, acyclic matchings and cell
complexes are collected for being used in the sequel.

###  2.1. Posets, matchings, and cell complexes

Let $P$ be a poset and $x,y\in P$ such that $x<y$. We say that $y$ covers $x$,
written as $x\lessdot y$, if there is no $z\in P$ such that $x<z<y$.

######  Definition 2.1.

Let $P$ be a poset with cover relation $\lessdot$. An involution
$M:P\rightarrow P$ (i.e., $M\circ M=id$) such that for each $x\in P$,
$x\lessdot M(x)$ or $M(x)\lessdot x$ or $M(x)=x$ is called a _matching_. An
element $x\in P$ such that $M(x)=x$ is called _critical_.

######  Definition 2.2.

Let $P$ be a poset and $M$ be a matching on $P$. Consider the Hasse diagram of
$P$ as a directed graph with edges $x\leftarrow y$ if $x\lessdot y$ for
$x,y\in P$. From the Hasse diagram of $P$ construct a new directed graph
$G_{M}(P)$ by reversing each arrow $x\leftarrow y$ to $x\rightarrow y$ if
$y=M(x)$. The matching $M$ on $P$ is called _acyclic_ if there are no directed
cycles in $G_{M}(P)$.

In $G_{M}(P)$, the unchanged edge $x\leftarrow y$ is said to be _downward_
while the edge $x\rightarrow M(x)$ is said to be _upward_. Since for any
matching $M$, no two incident edges are directed upward in $G_{M}(P)$, the
next lemma follows:

######  Lemma 2.3.

If a matching $M$ on a poset $P$ is not acyclic, then any directed cycle in
$G_{M}(P)$ is of the form

$a_{1}\lessdot M(a_{1})\gtrdot a_{2}\lessdot M(a_{2})\ldots\gtrdot
a_{k}\lessdot M(a_{k})\gtrdot a_{1},$

for pairwise distinct $a_{i}$ and some $k\geq 2$.

Let $\Delta$ be a finite regular cell complex. The _face poset_ of $\Delta$,
denoted by $P(\Delta)$, is the poset of all cells ordered by set inclusion of
their closures where a minimum element (sometimes thought of as the empty
cell) is also in $P(\Delta)$. For more on regular cell complexes, see [13].

######  Definition 2.4.

Let $\Delta$ be a finite regular cell complex and $P(\Delta)$ its face poset.
Then $\Delta$ is called a _boolean cell complex_ if $P(\Delta)$ is a
_simplicial_ poset (i.e., a poset with a minimum element in which every
principal order ideal is isomorphic to a boolean lattice).

Conversely, any finite simplicial poset is the face poset of a boolean cell
complex which is unique (up to cellular isomorphism).

The following is one of the most useful consequences of discrete Morse theory.
It will serve as our main topological tool.

######  Theorem 2.5.

[3, Theorem 6.3 ] Let $\Delta$ be a boolean cell complex and let $M$ be an
acyclic matching on the face poset $P(\Delta)$. If there are $c$ critical
cells, all of the same dimension $m$, then $\Delta$ is homotopy equivalent to
a wedge of $c$ spheres of dimension $m$.

######  Corollary 2.6.

If an acyclic matching $M$ on $P(\Delta)$ does not have any critical cells,
then $\Delta$ is contractible.

Lemma 2.7 has been discovered by Jonsson [11] and independently by Hersh [6].
It provides a way to patch together acyclic matchings.

######  Lemma 2.7.

Let $\Delta$ be a boolean cell complex such that $\Delta=\bigcup_{\alpha\in
Q}\Delta_{\alpha}$ where the index set $Q$ is partially ordered. Assume that:

  1. (1)

Each cell $\gamma\in\Delta$ is only in one $\Delta_{\alpha}$ (i.e., the union
is disjoint).

  2. (2)

$\bigcup_{\beta\leq\alpha\in Q}\Delta_{\beta}$ is an order ideal of
$P(\Delta)$ for every $\alpha\in Q$.

For each $\alpha\in Q$, suppose that $M_{\alpha}$ is an acyclic matching on
$P(\Delta_{\alpha})$ where $P(\Delta_{\alpha})$ is the subposet of $P(\Delta)$
induced by $\Delta_{\alpha}$. Then $\bigcup_{\alpha\in Q}M_{\alpha}$ is an
acyclic matching on $P(\Delta)$.

###  2.2. Coxeter systems

We review some facts on Coxeter groups for later use. For more, we refer the
reader to [1] or [10].

Let $W$ be a group generated by a finite set of involutions $S$, where
$(ss^{\prime})^{m(s,s^{\prime})}=e$ for $m(s,s)=1$,
$m(s,s^{\prime})=m(s^{\prime},s)\geq 2$ for all $s\neq s^{\prime}$ in $S$ and
$m(s,s^{\prime})\in\mathbb{Z}^{+}\cup\\{\infty\\}$. Here, $e$ denotes the
identity element in $W$. The group $W$ as described above, together with its
generating set $S$ form a pair $(W,S)$ which is called a _Coxeter system_. Any
$w\in W$ is a product of generators in $S$. That is $w=s_{1}s_{2}\cdots s_{k}$
for some $s_{i}\in S$. If $k$ is smallest among all such expressions for $w$,
then $k$ is called the _length_ of $w$, denoted $\ell(w)=k$, and
$s_{1}s_{2}\cdots s_{k}$ is called a _reduced expression_ for $w$. Note that
we do not distinguish notationally between an expression and the group element
it represents.

A Coxeter system $(W,S)$ is sometimes represented by its _Coxeter graph_. It
is the edge-labeled (simple, undirected) graph with vertex set $S$ and an edge
labeled $m(s,s^{\prime})$ connecting two vertices $s$ and $s^{\prime}$ if
$m(s,s^{\prime})\geq 3$. By convention, the label is usually omitted if it is
equal to 3, see Figure 1.

If $W$ is finite, it has a unique longest element $w_{0}$. A subgroup $W_{J}$
of $W$, generated by $J\subseteq S$ is called _parabolic_. If $W_{J}$ is
finite, its longest element is denoted by $w_{0}(J)$.

A Coxeter system is _irreducible_ if its Coxeter graph is connected. The
Coxeter graphs in Figure 1 indicate the _classification_ of finite irreducible
Coxeter groups. There are three classical families of types $A_{n}$ $(n\geq
1)$, $B_{n}$ $(n\geq 2)$, $D_{n}$ $(n\geq 4)$, six exceptional groups of types
$E_{6}$, $E_{7}$, $E_{8}$, $F_{4}$, $H_{3}$ and $H_{4}$, and one family of
dihedral groups of type $I_{2}(m)$ $(m\geq 3)$.

Figure 1. The finite, irreducible Coxeter groups.

Given $w\in W$, define the _right descent_ set of $w$ by

$D_{R}(w)=\\{s\in S|\ell(ws)<\ell(w)\\}.$

For $s\in S$, $s\in D_{R}(w)$ if and only if $w$ has a reduced expression that
ends in $s$.

Let $T:=\\{wsw^{-1}|w\in W,s\in S\\}$ be the set of _reflections_ in $W$.

######  Definition 2.8.

For $u,w\in W$, let $u\rightarrow w$ if there is $t\in T$ such that $w=ut$ and
$\ell(u)<\ell(w)$. The _Bruhat order_ is the partial order relation on $W$
which is defined by $u\leq w$ if there is a sequence

$u=u_{0}\rightarrow u_{1}\rightarrow\cdots u\rightarrow u_{n}=w.$

######  Definition 2.9.

Let $(W,S)$ be a Coxeter system and consider $W$ as a poset under Bruhat
order. An element $w\in W$ is called _boolean_ if its lower principal order
ideal is isomorphic to a boolean algebra.

The following lemma is a characterization of boolean elements. For $W$ of type
$A$, $B$, or $D$ this lemma is [18, Proposition 7.3].

######  Lemma 2.10.

[15] Let $(W,S)$ be a Coxeter system and $w\in W$. Then $w$ is boolean if and
only if it has some reduced expression without repeated letters.

In fact, Lemma 2.10 is equivalent to the statement that “$w$ is boolean if and
only if _no_ reduced expression for $w$ has repeated letters.” Let
$\mathbb{B}(W)$ denote the subposet of $W$ induced by the boolean elements.
The subposet $\mathbb{B}(W)$ is the _boolean ideal_. Note that $\mathbb{B}(W)$
is a simplicial poset.

######  Definition 2.11.

For a Coxeter system $(W,S)$, the _boolean complex_ of $(W,S)$ is the boolean
cell complex $\Delta(W)$ whose face poset is $\mathbb{B}(W)$.

All maximal cells of $\Delta(W)$ have dimension $|S|-1$. Boolean complexes of
Coxeter groups were studied by Ragnarsson and Tenner [15]. They showed that
for every Coxeter system $(W,S)$, there is a nonnegative integer $\beta(W)$
called _boolean number_ such that $\Delta(W)$ is homotopy equivalent to a
wedge of $\beta(W)$ spheres of dimension $|S|-1$. Moreover, they found a
recurrence formula for computing $\beta(W)$ in terms of the (unlabeled)
Coxeter graph. Note that the labels of the Coxeter graph have no effect in
determining $\beta(W)$. That is, if $W$ and $W^{\prime}$ have isomorphic
Coxeter graphs when ignoring all edge labels that are at least 4, then
$\mathbb{B}(W)\cong\mathbb{B}(W^{\prime})$. In particular,
$\beta(W)=\beta(W^{\prime})$ [15].

###  2.3. Reduced $\underline{S}$-expressions

In this subsection we review some material on reduced
$\underline{S}$-expressions and boolean involutions needed in Sections 3 and
4.

For a Coxeter system $(W,S)$, let $I=\\{w\in W|w=w^{-1}\\}$ denote the set of
involutions. Consider an alphabet $\underline{S}=\\{\underline{s}\>|\>s\in
S\\}$.

######  Definition 2.12.

The free monoid $\underline{S}^{\ast}$ acts from the right on the set $W$ by

$w\underline{s}=\begin{cases}ws\>\>\>\>\text{if}\>\>\>sws=w,\\\
sws\>\>\>\text{otherwise},\end{cases}$

and
$w\underline{s}_{1}\underline{s}_{2}\cdots\underline{s}_{k}=(\cdots((w\underline{s}_{1})\underline{s}_{2})\cdots\underline{s}_{k})$.

Note that $w\underline{s}\,\underline{s}=w$ for all $w\in W$ and $s\in S$. For
the identity element $e\in W$, write
$\underline{s}_{1}\underline{s}_{2}\cdots\underline{s}_{k}$ for
$e\underline{s}_{1}\underline{s}_{2}\cdots\underline{s}_{k}$.

For every $w\in I$ there are symbols
$\underline{s}_{1},\underline{s}_{2},\ldots,\underline{s}_{k}$ such that
$w=\underline{s}_{1}\underline{s}_{2}\cdots\underline{s}_{k}$ and, conversely,
all such elements are involutions [16]. The minimal $k$ such that
$w=\underline{s}_{1}\underline{s}_{2}\cdots\underline{s}_{k}$ for some
$\underline{s}_{1},\underline{s}_{2},\ldots,\underline{s}_{k}\in\underline{S}$
is called the $rank$ of $w$, and is denoted $\rho(w)$. The expression
$\underline{s}_{1}\underline{s}_{2}\cdots\underline{s}_{k}$ is then called a
reduced $\underline{S}$-expression for $w$.

Reduced $\underline{S}$-expressions appears under various names in the
literature. It is (the right-handed version of) what Richardson and Springer
call “admissible sequences” [16] and Hamaker, Marberg, and Pawlowski [4] refer
to as “involution words”. In this paper the notation is taken from [7, 8].

######  Definition 2.13.

Let
$\alpha_{\underline{s},\underline{s}^{\prime}}=\underline{s}\,\underline{s}^{\prime}\underline{s}\cdots$
be the alternating word in $\underline{s}$ and $\underline{s}^{\prime}$ where
the number of letters $\underline{s}$ and $\underline{s}^{\prime}$ in
$\alpha_{\underline{s},\underline{s}^{\prime}}$ is equal to $m(s,s^{\prime})$.
Operating on words in $\underline{S}^{\ast}$, we define a _braid move_ as the
replacement of $\alpha_{\underline{s},\underline{s}^{\prime}}$ by
$\alpha_{\underline{s}^{\prime},\underline{s}}$.

In fact braid moves in a reduced $\underline{S}$-expression for $w\in I$
preserve $w$ (see [4]).

######  Definition 2.14.

Let $\mathcal{\hat{R}}(w)$ denote the set of all reduced
$\underline{S}$-expressions for $w\in I$. For $u,w\in I$, the replacement of
one element in $\mathcal{\hat{R}}(u)$ by another in the beginning of a reduced
$\underline{S}$-expression for $w\in I$ is called an _initial move_.

The following theorem provides a minimal list of initial moves that combined
with braid moves suffice to connect all elements of $\mathcal{\hat{R}}(w)$.

######  Theorem 2.15.

[5] Let $(W,S)$ be a Coxeter system and $w\in I$. Any two reduced
$\underline{S}$-expressions for $w$ can be connected by a sequence of braid
moves and initial moves that replace $u$ with $v$ if
$u,v\in\mathcal{\hat{R}}(w_{0}(J))$ for some $J\subseteq S$. The following
parabolic subgroups $W_{J}$ are necessary and sufficient:

  1. (1)

$W_{J}$ of type $B_{3}$;

  2. (2)

$W_{J}$ of type $D_{4}$;

  3. (3)

$W_{J}$ of type $H_{3}$;

  4. (4)

$W_{J}$ of type $I_{2}(m)$, for $m\geq 3$.

In other words, given $x,y\in\mathcal{\hat{R}}(w)$, there exist some
$x_{0},x_{1},\cdots,x_{k}$ such that $x=x_{0},x_{k}=y$ and for all $i$,
$x_{i}$ differs from $x_{i+1}$ by a braid move or an initial move of the kind
specified in Theorem 2.15.

######  Lemma 2.16.

[7] Let $w\in I$. Then $s\in D_{R}(w)$ if and only if $w$ has a reduced
$\underline{S}$-expression that ends in $\underline{s}$. Moreover,
$\rho(w\underline{s})=\rho(w)+1$ if and only if $s\not\in D_{R}(w)$.

Let $\mathrm{Br}(I)$ denote the subposet of the Bruhat order on $W$ induced by
$I$. The order relation on $\mathrm{Br}(I)$ has been characterized in [7, 16,
17].

######  Lemma 2.17 (Subword property of $I$).

[7] Suppose that $v,w\in I$ and
$\underline{s}_{1}\underline{s}_{2}\cdots\underline{s}_{k}$ is a reduced
$\underline{S}$-expression for $w$. Then $v\leq w$ if and only if
$v=\underline{s}_{i_{1}}\underline{s}_{i_{2}}\cdots\underline{s}_{i_{r}}$ for
some $1\leq i_{1}\leq i_{2}\leq\cdots\leq i_{r}\leq k$.

For $w\in I$, let $B(w)$ denote the principal order ideal generated by $w$ in
$\mathrm{Br}(I)$. Then we have:

######  Definition 2.18.

An element $w\in I$ is called a _boolean involution_ if $B(w)$ is isomorphic
to a boolean algebra.

Analogously to the characterization of boolean elements (i.e., Theorem 2.10),
we have:

######  Proposition 2.19.

[9] Let $w\in I$. Then the following are equivalent:

  1. (1)

$w$ is a boolean involution.

  2. (2)

$w$ has some reduced $\underline{S}$-expression without repeated letters.

In fact, Proposition 2.19 is equivalent to “$w$ is a boolean involution if and
only if _no_ reduced $\underline{S}$-expression for $w$ has repeated letters.”

Suppose $s,s^{\prime}\in S$ satisfy $m(s,s^{\prime})=3$. Let us say that a
_half-braid move_ is the replacement of of
$\underline{s}\,\underline{s}^{\prime}$ by
$\underline{s}^{\prime}\underline{s}$ in the beginning of a reduced
$\underline{S}$-expression.

######  Proposition 2.20.

Any two reduced $\underline{S}$-expressions for a boolean involution $w\in I$
can be connected by a sequence of braid moves and half-braid moves.

###### Proof.

In the list of parabolic subgroups given in Theorem 2.15, it is only in type
$I_{2}(3)$ that the longest element is a boolean involution. It has only two
reduced $\underline{S}$-expressions, and the initial move replacing one of
them by the other is a half-braid move. ∎

Let $P(\Delta_{\mathrm{inv}}(W))$ denote the subposet of $\mathrm{Br}(I)$
induced by all boolean involutions. It is a simplicial poset which we call the
_boolean involution ideal_. We have the following definition:

######  Definition 2.21.

Let $\Delta_{\mathrm{inv}}(W)$ be the boolean cell complex whose face poset is
$P(\Delta_{\mathrm{inv}}(W))$.

By abuse of notation, we identify a boolean involution with the cell it
represents. It follows from Proposition 2.20 together with Lemma 2.16 that
every $\underline{S}$-expression without repeated letters is reduced. Hence,
every maximal cell in $\Delta_{\mathrm{inv}}(W)$ has dimension $|S|-1$. Let
$\beta_{I}(W)$ denote the absolute value of the reduced Euler characteristic
of $\Delta_{\mathrm{inv}}(W)$.

In contrast to the situation for the boolean complex $\Delta(W)$, it is not
the case that $\Delta_{\mathrm{inv}}(W)$ only depends on the unlabeled Coxeter
graph. As the following lemma shows, it is however not necessary to keep track
of the actual labels:

######  Lemma 2.22.

If the Coxeter graph of a Coxeter system $(W^{\prime},S)$ is obtained from
that of another system $(W,S)$ by increasing some edge labels which are at
least $4$, then
$\Delta_{\mathrm{inv}}(W^{\prime})\cong\Delta_{\mathrm{inv}}(W)$.

###### Proof.

In a reduced $\underline{S}$-expression without repeated letters, the possible
braid moves are provided by the vertices that are not adjacent in the Coxeter
graph, and the half-braid moves come from the vertices that are connected by
an edge labeled $3$ (i.e., with omitted label). It follows that the boolean
involution ideals of $W$ and $W^{\prime}$ are isomorphic. ∎

##  3\.  Acyclic matchings on the boolean involution ideal

Say that a Coxeter system $(W,S)$ (with $S$ finite) is ordered if the
generating set $S$ is endowed with a total order. For such a system, with
$s=\max S$, define

$P_{1}(W):=\\{a\in\Delta_{\mathrm{inv}}(W)\mid\ \underline{s}\not\leq a\text{
or }s\in D_{R}(a)\\}.$

Then, let

$\begin{split}\Gamma(W):&=\Delta_{\mathrm{inv}}(W)\setminus P_{1}(W)\\\
&=\\{a\in\Delta_{\mathrm{inv}}(W)\mid\underline{s}\leq a\text{ and }s\not\in
D_{R}(a)\\}.\end{split}$

######  Definition 3.1.

Suppose $(W,S)$ is an ordered Coxeter system. A matching $M$ on
$P(\Delta_{\mathrm{inv}}(W))$ is a $\Gamma$-matching if it satisfies the
following properties:

  * •

$M$ is acyclic;

  * •

All critical cells of $M$ are of top dimension $|S|-1$;

  * •

$M$ preserves $\Gamma(W)$;

  * •

All critical cells of $M$ belong to $\Gamma(W)$.

In particular, by Theorem 2.5, if $P(\Delta_{\mathrm{inv}}(W))$ has a
$\Gamma$-matching, then $\Delta_{\mathrm{inv}}(W)$ is homotopy equivalent to a
wedge of $\beta_{I}(W)$ spheres of dimension $|S|-1$.

If $(W,S)$ is ordered, we let $W_{-k}$ denote the parabolic subgroup which is
generated by the first $|S|-k$ elements of $S$, i.e. the $k$ largest elements
are removed from $S$. The corresponding subsystem is ordered with the order
inherited from $S$.

######  Definition 3.2.

Let $(W,S)$ be an ordered Coxeter system with $S=\\{s_{1}<\cdots<s_{n}\\}$
where $n\geq 3$. We say that $(W,S)$ is path ended if $s_{n}$ commutes with
every $s_{i}$ for $i\leq n-2$, $s_{n-1}$ commutes with every $s_{i}$ for
$i\leq n-3$, $s_{n}s_{n-1}s_{n}=s_{n-1}s_{n}s_{n-1}$ and
$s_{n-1}s_{n-2}s_{n-1}=s_{n-2}s_{n-1}s_{n-2}$.

Definition 3.2 means that the Coxeter graph of a path ended system $(W,S)$ is
as illustrated in Figure 2, where the parabolic subgroups $W_{-2}$ and
$W_{-3}$ are also indicated; $W_{-2}$ is generated by all $s_{i}$ inside the
blue circle, and $W_{-3}$ is generated by all $s_{i}$ inside the black circle,
respectively.

$\;\;\;\;\;\;\;\;\;\;s_{n-2}$$s_{n-1}$$s_{n}$$W_{-2}$$W_{-3}$ Figure 2. A
Coxeter graph of a path ended system $(W,S)$

Now we have the following theorem.

######  Theorem 3.3.

Let $(W,S)$ be a path ended Coxeter system. Assume that
$P(\Delta_{\mathrm{inv}}(W_{-2}))$ and $P(\Delta_{\mathrm{inv}}(W_{-3}))$ have
$\Gamma$-matchings $M^{-2}$ and $M^{-3}$, respectively. Then,
$P(\Delta_{\mathrm{inv}}(W))$ has a $\Gamma$-matching $M$. Moreover, the
number of critical elements of $M$ equals the number of critical elements of
$M^{-2}$ plus the number of critical elements of $M^{-3}$.

The proof of Theorem 3.3 is deferred to the next subsection. Here we proceed
to apply it in order to prove Theorem 3.4 below, which is the main result of
this paper.

For two ordered Coxeter systems $(W,S)$ and $(Z,S^{\prime})$, we say that
$(Z,S^{\prime})$ is a path extension of $(W,S)$ if
$S=\\{s_{1}<\cdots<s_{n}\\}$ and $S^{\prime}=\\{s_{1}<\cdots<s_{N}\\}$ for
some $N\geq n$, and for all $1\leq i\leq N$ and $n<j\leq N$ with $i\neq j$, it
holds that

$m(s_{i},s_{j})=\begin{cases}2&\text{if $|i-j|\geq 2$,}\\\ 3&\text{if
$|i-j|=1$.}\end{cases}$

In other words, we obtain the Coxeter graph of a path extension of $(W,S)$
from that of $(W,S)$ by attaching a (possibly empty) path at the maximum
vertex $s_{n}$. Figure 3 illustrates this situation for a path ended system
$(W,S)$. This is the setting of the upcoming theorem.

$\;\;\;\;\;\;\;\;s_{{}_{n-2}}$$\;\;\;\;\;s_{{}_{n-1}}$$s_{{}_{n}}$$s_{{}_{N}}$$W$
Figure 3. A Coxeter system $(Z,S^{\prime})$ as a path extension of a path
ended Coxeter system $(W,S)$.

######  Theorem 3.4.

Let $(W,S)$ be a path ended Coxeter system such that
$P(\Delta_{\mathrm{inv}}(W_{-1}))$, $P(\Delta_{\mathrm{inv}}(W_{-2}))$ and
$P(\Delta_{\mathrm{inv}}(W_{-3}))$ all have $\Gamma$-matchings. Then, for
every path extension $(Z,S^{\prime})$ of $(W,S)$, $\Delta_{\mathrm{inv}}(Z)$
is homotopy equivalent to a wedge of $(|S^{\prime}|-1)$-dimensional spheres.
The number of spheres satisfies the recurrence
$\beta_{I}(Z)=\beta_{I}(Z_{-2})+\beta(Z_{-3}).$

###### Proof.

Since $(Z,S^{\prime})$ is path ended, Theorem 3.3 together with Theorem 2.5
provide the desired conclusions if we are able to show that $Z_{-2}$ and
$Z_{-3}$ have $\Gamma$-matchings. By induction on $|S^{\prime}|$, it however
follows from Theorem 3.3 that $Z_{-1},Z_{-2}$, and $Z_{-3}$ all have
$\Gamma$-matchings. ∎

###  3.1. Proof of Theorem 3.3

We now proceed to prove Theorem 3.3. First let us establish a couple of
lemmas.

Let $(W,S)$ be as in Definition 3.2.

######  Lemma 3.5.

Let $w\in W$ be a boolean involution. If $\underline{s}_{n-1}$ comes after
both $\underline{s}_{n-2}$ and $\underline{s}_{n}$ in some reduced
$\underline{S}$-expression for $w$, then this holds in every reduced
$\underline{S}$-expression for $w$.

###### Proof.

Note that if $\underline{s}_{n-1}$ comes after both $\underline{s}_{n-2}$ and
$\underline{s}_{n}$ in a reduced $\underline{S}$-expression for $w$, then this
cannot be changed by a half-braid move or a braid move. Hence the result
follows from Proposition 2.20.

∎

Now define

$P_{2}(W):=\\{x\underline{s}_{n-2}\underline{s}_{n}\underline{s}_{n-1}:x\in\Delta_{\mathrm{inv}}(W_{-3})\\}.$

######  Lemma 3.6.

We have $P_{2}(W)\subseteq\Gamma(W)$.

###### Proof.

By Lemma 3.5, $\underline{s}_{n-1}$ comes after $\underline{s}_{n-2}$ and
$\underline{s}_{n}$ in every reduced $\underline{S}$-expression for every
element $a\in P_{2}(W)$. Hence by Lemma 2.16, $s_{n}\not\in D_{R}(a)$. ∎

Define $P_{3}(W):=\Gamma(W)\backslash P_{2}(W)$. Note that $P_{1}(W)$,
$P_{2}(W)$, and $P_{3}(W)$ are pairwise disjoint sets whose union is
$\Delta_{\mathrm{inv}}(W)$.

######  Lemma 3.7.

$P_{3}(W)=\\{y\underline{s}_{n}\underline{s}_{n-1}:y\in\Gamma(W_{-2})\\}$.

###### Proof.

For any $a\in\\{y\underline{s}_{n}\underline{s}_{n-1}:y\in\Gamma(W_{-2})\\}$
we have that $a\not\in P_{2}(W)$ since $s_{n-2}\not\in D_{R}(y)$. By Lemma
3.5, $\underline{s}_{n-1}$ comes after $\underline{s}_{n-2}$ and
$\underline{s}_{n}$ in every reduced $\underline{S}$-expression for $a$, hence
$\underline{s}_{n}\leq a$ and $s_{n}\notin D_{R}(a)$. Thus
$a\in\Gamma(W)\backslash P_{2}(W)=P_{3}(W)$.

Conversely, assume that $a\in P_{3}(W)$. Since $\underline{s}_{n}\leq a$ and
$s_{n}\not\in D_{R}(a)$, $\underline{s}_{n-1}$ comes after $\underline{s}_{n}$
in every reduced $\underline{S}$-expression for $a$. Hence, no reduced
$\underline{S}$-expression for $a$ can begin with
$\underline{s}_{n}\underline{s}_{n-1}$, which means that $\underline{s}_{n-1}$
also comes after $\underline{s}_{n-2}$ in every reduced
$\underline{S}$-expression for $a$. Therefore,
$a=y\underline{s}_{n}\underline{s}_{n-1}$ for some
$y\in\Delta_{\mathrm{inv}}(W_{-2})$. Moreover, $y\in\Gamma(W_{-2})$ because
$a\not\in P_{2}(W)$. ∎

###### Proof of Theorem 3.3.

Define matchings $M_{1},M_{2}$ and $M_{3}$ on $P_{1}(W),P_{2}(W)$ and
$P_{3}(W)$ respectively as:

  1. (1)

$M_{1}(a):=a\underline{s}_{n}$ for $a\in P_{1}(W)$;

  2. (2)

$M_{2}(a):=M^{-3}(x)\underline{s}_{n-2}\underline{s}_{n}\underline{s}_{n-1}$
for $a=x\underline{s}_{n-2}\underline{s}_{n}\underline{s}_{n-1}\in P_{2}(W)$;

  3. (3)

$M_{3}(a):=M^{-2}(y)\underline{s}_{n}\underline{s}_{n-1}$ for
$a=y\underline{s}_{n}\underline{s}_{n-1}\in P_{3}(W)$.

Note that $M^{-2}(y)\in\Gamma(W_{-2})$ since $M^{-2}$ preserves
$\Gamma(W_{-2})$. This ensures that $M_{3}$ is well defined.

Define also a matching

$M(a):=\begin{cases}M_{1}(a)\>\text{if}\>a\in P_{1}(W),\\\
M_{2}(a)\>\text{if}\>a\in P_{2}(W),\\\ M_{3}(a)\>\text{if}\>a\in
P_{3}(W)\end{cases}$

on $P(\Delta_{\mathrm{inv}}(W))$. We shall show that $M$ is a
$\Gamma$-matching on $P(\Delta_{\mathrm{inv}}(W))$.

A. $M$ is an acyclic matching on $P(\Delta_{\mathrm{inv}}(W))$:

We will first show that the matchings $M_{1}$, $M_{2}$ and $M_{3}$ are acyclic
on $P_{1}(W)$, $P_{2}(W)$ and $P_{3}(W)$ respectively and then use Lemma 2.7
to show that $M$ is an acyclic matching on $P(\Delta_{\mathrm{inv}}(W))$.

The matching $M_{1}$ is acyclic. To see it: By Lemma 2.3, if $M_{1}$ has a
cycle on $P_{1}(W)$, then $a_{0}\lessdot
M_{1}(a_{0})=a_{0}\underline{s}_{n}\gtrdot a_{1}$ and $a_{1}\lessdot
M_{1}(a_{1})=a_{1}\underline{s}_{n}$ for some $a_{0}\neq a_{1}$. However since
$\underline{s}_{n}\nleq a_{0}$ and $\underline{s}_{n}\nleq a_{1}$ we have that
$a_{0}=a_{1}$, so there is no cycle. Hence $M_{1}$ is acyclic.

Define a map $g:P(\Delta_{\mathrm{inv}}(W_{-3}))\rightarrow P_{2}(W)$ by
$g(x)=x\underline{s}_{n-2}\underline{s}_{n}\underline{s}_{n-1}$. The subword
property implies that $g$ is a poset isomorphism. Since $M^{-3}$ is acyclic,
$M_{2}$ is an acyclic matching.

Similarly, $\Gamma(W_{-2})$ and $P_{3}(W)$ are isomorphic. Since $M^{-2}$ is
acyclic and $M^{-2}$ preserves $\Gamma(W_{-2})$, $M_{3}$ is an acyclic
matching.

Our goal is to show that $M$ is acyclic using Lemma 2.7. Let $Q$ be the
(total) order whose Hasse diagram is as below:

$P_{1}(W)$$P_{2}(W)$$P_{3}(W)$ Figure 4. Hasse diagram of $Q$.

We proceed to verify the hypotheses of the lemma.

  1. (1)

As we have already seen, each cell belongs to only one $P_{i}(W)$ for
$i=1,2,3$.

  2. (2)

We show that $P_{1}(W)$ and $P_{1}(W)\cup P_{2}(W)$ are order ideals. To
observe that $P_{1}(W)$ is an order ideal we let $a\in P_{1}(W)$ and $t\leq
a$. If $\underline{s}_{n}\nleq a$, then $\underline{s}_{n}\nleq t$ and hence
$t\in P_{1}(W)$. If $s_{n}\in D_{R}(a)$, then either $\underline{s}_{n}\nleq
t$ or $s_{n}\in D_{R}(t)$. So $t\in P_{1}(W)$. Thus $P_{1}(W)$ is an order
ideal.

Also $P_{1}(W)\cup P_{2}(W)$ is an order ideal. If $a\in P_{2}(W)$ and $t<a$,
then we have $t<a=x\underline{s}_{n-2}\underline{s}_{n}\underline{s}_{n-1}$.
If $\underline{s}_{n-2}\underline{s}_{n}\underline{s}_{n-1}\leq t$ then $t\in
P_{2}(W)$. If $\underline{s}_{n-2}\underline{s}_{n}\underline{s}_{n-1}\nleq t$
then $\underline{s}_{i}\nleq t$ for at least one of $i=n-2,n-1,n$. If
$\underline{s}_{n-2}\nleq t$ then
$t=x\underline{s}_{n}\underline{s}_{n-1}=x\underline{s}_{n-1}\underline{s}_{n}\in
P_{1}(W)$. Similarly if $\underline{s}_{n}$ or $\underline{s}_{n-1}\nleq t$,
then $t=x\underline{s}_{n-2}\underline{s}_{n-1}\in P_{1}(W)$ or
$t=x\underline{s}_{n-2}\underline{s}_{n}\in P_{1}(W)$. Hence $P_{1}(W)\cup
P_{2}(W)$ is an order ideal.

Now since $P_{1}(W),P_{1}(W)\cup P_{2}(W)$ and $P_{1}(W)\cup P_{2}(W)\cup
P_{3}(W)$ are order ideals, and $M_{1},M_{2}$ and $M_{3}$ are acyclic
matchings on $P_{1}(W),P_{2}(W)$, and $P_{3}(W)$ respectively, then by Lemma
2.7, $M$ is an acyclic matching on $P(\Delta_{\mathrm{inv}}(W))$.
B. All critical cells are of top dimension $n-1$: Let $a$ be a critical cell
of $M$. Since $M_{1}$ is complete on $P_{1}(W)$, then either
$a=x\underline{s}_{n-2}\underline{s}_{n}\underline{s}_{n-1}$ with $x$ a
critical cell of $M^{-3}$ or $a=y\underline{s}_{n}\underline{s}_{n-1}$ with
$y$ a critical cell of $M^{-2}$. By the assumptions on $M^{-2}$ and $M^{-3}$,
$\text{dim}(a)=n-1$ in both cases.
C. $M$ preserves $\Gamma(W)$: That the matching $M$ preserves $\Gamma(W)$
follows by construction since $\Gamma(W)=P_{2}(W)\cup P_{3}(W)$.
D. All critical cells belong to $\Gamma(W)$: Since $M$ is complete on
$P_{1}(W)$, there are no critical cells in $P_{1}(W)$. So all critical cells
of $M$ are in $P(\Delta_{\mathrm{inv}}(W))\backslash P_{1}(W)=\Gamma(W)$.

This completes the proof that $M$ is a $\Gamma$-matching on
$P(\Delta_{\mathrm{inv}}(W))$. Finally, the assertion about the number of
critical cells is immediate from part B above. ∎

##  4\. Applications of Theorem 3.4

In this section we shall calculate the homotopy types of some explicit boolean
involution complexes, including all those of finite Coxeter groups. Barring
some small examples that are easily computed by hand, we achieve this by
identifying $\Gamma$-matchings of some suitably chosen ordered Coxeter
systems, and then invoking Theorem 3.4.

######  Theorem 4.1.

If $W$ is irreducible and finite, the homotopy type of
$\Delta_{\mathrm{inv}}(W)$ is as in the following table

_Coxeter group $W$_ | _Homotopy type of $\Delta_{\mathrm{inv}}(W)$_
---|---
$A_{n}$ | $\beta_{I}(A_{n})\cdot S^{n-1}$
$B_{n}$ | $\beta_{I}(B_{n})\cdot S^{n-1}$
$D_{n}$ | $\beta_{I}(D_{n})\cdot S^{n-1}$
$E_{6}$ | _a point_
$E_{7}$ | $S^{6}$
$E_{8}$ | $S^{7}$
$F_{4}$ | _a point_
$H_{3}$ | $S^{2}$
$H_{4}$ | $S^{3}$
$I_{2}(m)$ | $S^{1}$

where the Betti numbers in the classical types are determined by:

  * •

$\beta_{I}(A_{n})=\beta_{I}(A_{n-2})+\beta_{I}(A_{n-3})$ with
$\beta_{I}(A_{1})=0,\beta_{I}(A_{2})=0$ and $\beta_{I}(A_{3})=1$,

  * •

$\beta_{I}(B_{n})=\beta_{I}(B_{n-2})+\beta_{I}(B_{n-3})$ with
$\beta_{I}(B_{2})=1,\beta_{I}(B_{3})=1$ and $\beta_{I}(B_{4})=1$,

  * •

$\beta_{I}(D_{n})=\beta_{I}(D_{n-2})+\beta_{I}(D_{n-3})$ with
$\beta_{I}(D_{4})=1,\beta_{I}(D_{5})=1$ and $\beta_{I}(D_{6})=2$.

######  Remark 4.2.

Recall that the Padovan sequence $P_{0},P_{1},\ldots$ is determined by
$P_{0}=1,P_{1}=0,P_{2}=0$ and $P_{n}=P_{n-2}+P_{n-3}$ for $n\geq 3$. Hence,
$\beta_{I}(A_{n})=P_{n}$ for $n\geq 1$, $\beta_{I}(B_{n})=P_{n+3}$ for $n\geq
2$, and $\beta_{I}(D_{n})=P_{n+2}$ for $n\geq 2$. For more information about
the Padovan sequence see [14] and the references cited there.

Figure 5. Ordered Coxeter systems that appear in the proof of Theorem 4.1

###### Proof of Theorem 4.1.

We provide $\Gamma$-matchings of $P(\Delta_{\mathrm{inv}}(W))$. In types $A$,
$B$, $D$, and $E$, this is achieved by using Theorem 3.4. Somewhat nonstandard
type terminology will be employed in order to indicate specific ordered
Coxeter systems. Their Coxeter graphs are collected in Figure 5.

First, let us make a general observation. In order to verify that the boolean
involution ideal of a certain ordered Coxeter system $(W,S)$ has a
$\Gamma$-matching, it is enough to verify that the subposet induced by
$\Gamma(W)$ admits an acyclic matching with all critical cells of top
dimension. Namely, just as in the proof of Theorem 3.3, the subposet
$P_{1}(W)$ is an order ideal with a complete acyclic matching, and so the
union of these two matchings is a $\Gamma$-matching. This is illustrated for
$W$ of type $F_{4}$ in Figure 6. There, the entire acyclic matching on
$P(\Delta_{\mathrm{inv}}(W))$ is depicted. For the conclusion of the theorem,
it is however enough to inspect $\Gamma(W)$, which consists of the two yellow
elements.

Figure 6. A $\Gamma$-matching on $P(\Delta_{\mathrm{inv}}(F_{4}))$.

Type A. Consider $(W,S)$ of type $A_{n}$, $n\geq 4$, as an ordered Coxeter
system with the natural order on the generators as indicated in Figure 1.
Then, $(W,S)$ is a path extension of the path ended system $A_{4}$. In order
to obtain the desired result from Theorem 3.4, it therefore suffices to verify
the hypotheses, namely that the boolean involution ideals of $A_{3}$, $A_{2}$,
and $A_{1}$ have $\Gamma$-matchings. Concluding the proof in type $A$, one
readily verifies that
$\Gamma(A_{3})=\\{\underline{s}_{1}\underline{s}_{3}\underline{s}_{2}\\}$ and
$\Gamma(A_{2})=\Gamma(A_{1})=\emptyset$.

Type B. We proceed similarly to type $A$. This time, we begin with the ordered
type $B$ system in Figure 1 which, for $n\geq 4$, is a path extension of
$B_{4}$. Here,
$\Gamma(B_{3})=\\{\underline{s}_{1}\underline{s}_{3}\underline{s}_{2}\\}$,
$\Gamma(B_{2})=\\{\underline{s}_{2}\underline{s}_{1}\\}$, and
$\Gamma(B_{1})=\emptyset$.

Type D. Again, we argue in a similar way. Here, we begin with the ordered
system of type $D_{n}$ as in Figure 1, where $D_{n}$ is a path extension of
$D_{5}$ for all $n\geq 5$. The hypotheses are verified by observing that
$\Gamma(D_{2})=\emptyset$,
$\Gamma(D_{3})=\\{\underline{s}_{1}\underline{s}_{3}\underline{s}_{2}\\}$, and
$\Gamma(D_{4})=\\{\underline{s}_{1}\underline{s}_{4}\underline{s}_{3},$
$\underline{s}_{2}\underline{s}_{4}\underline{s}_{3},$
$\underline{s}_{1}\underline{s}_{4}\underline{s}_{3}\underline{s}_{2},$
$\underline{s}_{1}\underline{s}_{2}\underline{s}_{4}\underline{s}_{3},$
$\underline{s}_{2}\underline{s}_{4}\underline{s}_{3}\underline{s}_{1}\\}$.
Here, an acyclic matching on $\Gamma(D_{4})$ is depicted in Figure 7 where the
only critical cell is
$\underline{s}_{2}\underline{s}_{4}\underline{s}_{3}\underline{s}_{1}$.

Figure 7. An acyclic matching on $\Gamma(D_{4})$.

Type E. We proceed in a similar way as in types $A$, $B$, $D$ above. We start
with the ordered system of type $E$ as in Figure 1 where $E_{6}$, $E_{7}$, and
$E_{8}$ are path extensions of $E_{6}$. The hypotheses are readily checked by
observing that $\Gamma(E_{3})=\emptyset$,
$\Gamma(E_{4})=\\{\underline{s}_{2}\underline{s}_{4}\underline{s}_{3}$,
$\underline{s}_{1}\underline{s}_{4}\underline{s}_{2}$,
$\underline{s}_{1}\underline{s}_{2}\underline{s}_{4}\underline{s}_{3}$,
$\underline{s}_{1}\underline{s}_{4}\underline{s}_{2}\underline{s}_{3}\\}$ see
Figure 8, and $\Gamma(E_{5})$ is as depicted in Figure 9 where an acyclic
matching on $\Gamma(E_{5})$ with one critical cell,
$\underline{s}_{3}\underline{s}_{5}\underline{s}_{4}\underline{s}_{2}\underline{s}_{1}$,
is indicated.

Figure 8. An acyclic matching on $\Gamma(E_{4})$.

Figure 9. An acyclic matching on $\Gamma(E_{5})$.

Type H. The results in type $H$ are the same as in type $B$ by Lemma 2.22. ∎

Some concluding remarks are in order:

First, the previous theorem concerns irreducible groups, but reducible groups
are easily covered. Namely, taking products of Coxeter systems is readily seen
to imply taking the join of their corresponding boolean complexes of
involutions.

Second, there is of course nothing that prevents extending the approach of the
previous proof to path extensions which are not finite. For example,
$\tilde{F}_{4}$ is a path extension of $F_{4}$ and
$\Delta_{\mathrm{inv}}(\tilde{F}_{4})$ is homotopy equivalent to $S^{4}$,
$\tilde{E}_{8}$ is a path extension of $E_{8}$ and
$\Delta_{\mathrm{inv}}(\tilde{E}_{8})$ is homotopy equivalent to $S^{8}$, etc.

Third, just as we did for type $H$ in the proof above, we may apply Lemma 2.22
to obtain results for more Coxeter systems without additional effort.

Let us end with the natural question: Does the above result extend to all
Coxeter systems? That is, is $\Delta_{\mathrm{inv}}(W)$ homotopy equivalent to
a wedge of spheres of dimension $|S|-1$ for every system $(W,S)$?

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