A Multiparametric Quon Algebra

TL;DR

This paper introduces a generalized multiparametric quon algebra with deformed commutation relations, providing a mathematical framework for particle statistics that extends traditional quantum statistics.

Contribution

It generalizes the quon algebra by incorporating a matrix of deformation parameters and proves its realizability using hyperplane arrangement techniques.

Findings

The model is realizable for certain parameter values.

The generated module can be an indefinite Hilbert space.

Refines the extended Zagier's conjecture.

Abstract

The quon algebra is an approach to particle statistics introduced by Greenberg in order to provide a theory in which the Pauli exclusion principle and Bose statistics are violated by a small amount. We generalize these models by introducing a deformation of the quon algebra generated by a collection of operators $\mathtt{a}_i$, $i \in \mathbb{N}^*$ the set of positive integers, on an infinite dimensional module satisfying the $q_{i,j}$-mutator relations $\mathtt{a}_i \mathtt{a}_j^{\dag} - q_{i,j}\, \mathtt{a}_j^{\dag} \mathtt{a}_i = \delta_{i,j}$. The realizability of our model is proved by means of the Aguiar-Mahajan bilinear form on the chambers of hyperplane arrangements. We show that, for suitable values of $q_{i,j}$, the module generated by the particle states obtained by applying combinations of $\mathtt{a}_i$'s and $\mathtt{a}_i^{\dag}$'s to a vacuum state $|0\rangle$ is an indefinite Hilbert module. Furthermore, we refind the extended Zagier's conjecture established independently by Meljanac et al. and by Duchamp et al.

Equations

$${\mathtt{a}}_i{\mathtt{a}}_j^{†}=q_{i,j}{\mathtt{a}}_j^{†}{\mathtt{a}}_i+{\delta }_{i,j},$$

$${\mathtt{a}}_i|0⟩=0,$$

$$\tilde{P}:=\tilde{\mu }\prod _{s\in [n]}{\tilde{q}}_{i_s,j_s}\text{with}\tilde{\mu }=\bar{\mu },\text{and}{\tilde{q}}_{i_s,j_s}=q_{j_s,i_s},$$

$$\big{(}\mu\mathtt{a}|0\rangle,\,\nu\mathtt{b}|0\rangle\big{)}:=\mu\tilde{\nu}\,\langle 0|\mathtt{a}\,\mathtt{b}^{{\dagger}}|0\rangle\quad\text{with}\quad\langle 0|0\rangle=1,$$

$$\mathbf{B}:=\big{\{}\mathtt{a}_{i_{1}}^{{\dagger}}\dots\mathtt{a}_{i_{n}}^{{\dagger}}|0\rangle\ \big{|}\ (i_{1},\dots,i_{n})\in(\mathbb{N}^{*})^{n},\,n\in\mathbb{N}\big{\}}$$

$$\mathbf{M}=\bigoplus_{n\in\mathbb{N}^{*}}\bigoplus_{\displaystyle I\in\left(\!\!{\mathbb{N}^{*}\choose n}\!\!\right)}\mathbf{M}_{I}\quad\text{with}\quad\mathbf{M}_{I}=\Big{(}\langle 0|\,\mathtt{a}_{\dot{\tau}(n)}\dots\mathtt{a}_{\dot{\tau}(1)}\,\mathtt{a}_{\dot{\sigma}(1)}^{{\dagger}}\dots\mathtt{a}_{\dot{\sigma}(n)}^{{\dagger}}\,|0\rangle\Big{)}_{\dot{\sigma},\dot{\tau}\in\mathfrak{S}_{I}},$$

$${\mathbf{M}}_{[3]}=\left(\begin{array}{cccccc}1& q_{3,2}& q_{2,1}& q_{2,1}{q}_{3,1}& q_{3,1}{q}_{3,2}& q_{3,1}{q}_{2,1}{q}_{3,2}\\ q_{2,3}& 1& q_{2,1}{q}_{2,3}& q_{2,1}{q}_{3,1}{q}_{2,3}& q_{3,1}& q_{3,1}{q}_{2,1}\\ q_{1,2}& q_{1,2}{q}_{3,2}& 1& q_{3,1}& q_{3,2}{q}_{1,2}{q}_{3,1}& q_{3,2}{q}_{3,1}\\ q_{1,2}{q}_{1,3}& q_{1,2}{q}_{1,3}{q}_{3,2}& q_{1,3}& 1& q_{3,2}{q}_{1,2}& q_{3,2}\\ q_{1,3}{q}_{2,3}& q_{1,3}& q_{2,3}{q}_{2,1}{q}_{1,3}& q_{2,3}{q}_{2,1}& 1& q_{2,1}\\ q_{1,3}{q}_{1,2}{q}_{2,3}& q_{1,3}{q}_{1,2}& q_{2,3}{q}_{1,3}& q_{2,3}& q_{1,2}& 1\end{array}\right).$$

$$\det\mathbf{M}_{J}=\prod_{\begin{subarray}{c}K\in 2^{J}\\ \#K\,\geq 2\end{subarray}}\Big{(}1-\prod_{\{s,t\}\in\binom{K}{2}}q_{s,t}q_{t,s}\Big{)}^{(\#K\,-2)!\,(n-\,\#K\,+1)!}.$$

$$F:=\bigcap _{H\in \mathcal{A}}{H}^{{ϵ}_H(F)}\text{with}{ϵ}_H(F)\in \{+,0,-\}.$$

$$\mathrm{v}(C,C)=1\text{and}\mathrm{v}(C,D)=\prod _{H^{\epsilon }\in H_{C,D}}{h}_H^{\epsilon }\text{if}C\ne D.$$

$${\mathcal{A}}_n:=\{H_{i,j}|i,j\in [n],i<j\}\text{with}{H}_{i,j}=\{x\in {\mathbb{R}}^n|x_i=x_j\}.$$

$$C_{{\mathcal{A}}_n}=\{C_{\sigma }|\sigma \in {\mathfrak{S}}_n\}\text{with}{C}_{\sigma }:=\{x\in {\mathbb{R}}^n|x_{\sigma (1)}<x_{\sigma (2)}<\cdots <x_{\sigma (n)}\}.$$

$${\mathrm{v}}_n(C_{\sigma },C_{\tau })=\prod _{\begin{array}{c}\{i,j\}\in \left(\genfrac{}{}{0ex}{}{[n]}{2}\right)\\ i<j\\ {\tau }^{-1}\circ \sigma (i)>{\tau }^{-1}\circ \sigma (j)\end{array}}{q}_{\sigma (i),\sigma (j)},$$

$${\gamma }_n(C_{\tau }):=\sum _{\sigma \in {\mathfrak{S}}_n}{\mathrm{v}}_n(C_{\tau },C_{\sigma })C_{\sigma }.$$

$$\det\gamma_{n}=\prod_{\begin{subarray}{c}I\in 2^{[n]}\\ \#I\,\geq 2\end{subarray}}\Big{(}1-\prod_{\{i,j\}\in\binom{I}{2}}q_{i,j}q_{j,i}\Big{)}^{(\#I\,-2)!\,(n-\,\#I\,+1)!}.$$

$$\det \gamma =\prod _{E\in L_{\mathcal{A}}}(1-{\mathrm{b}}_E{)}^{{\beta }_E}.$$

$$L_{\mathcal{A}_{n}}^{\prime}=\big{\{}E_{I}\ \big{|}\ I\subseteq[n],\,|I|\geq 2\big{\}}\quad\text{and}\quad\beta_{E_{I}}=(\#I\,-2)!\,(n-\,\#I\,+1)!.$$

$$M_{\mathcal{A}_{n}}^{\lambda}:=\Big{\{}\sum_{\dot{\sigma}\in\mathfrak{S}_{I_{\lambda}}}x_{C_{\dot{\sigma}}}C_{\dot{\sigma}}\ \Big{|}\ x_{C_{\dot{\sigma}}}\in R_{\mathcal{A}_{n}}\Big{\}}.$$

$${\mathrm{v}}_n(C_{\dot{\tau }},C_{\upsilon \sigma })$$

$$=\sum _{\phi \in {\mathfrak{S}}_{\lambda }}\prod _{\begin{array}{c}\{i,j\}\in \left(\genfrac{}{}{0ex}{}{[n]}{2}\right)\\ i<j\\ {\sigma }^{-1}{\upsilon }^{-1}\circ \phi \tau (i)>{\sigma }^{-1}{\upsilon }^{-1}\circ \phi \tau (j)\end{array}}{q}_{\dot{\phi \tau (i)},\dot{\phi \tau (j)}}$$

$$=\sum _{\phi \in {\mathfrak{S}}_{\lambda }}\prod _{\begin{array}{c}\{i,j\}\in \left(\genfrac{}{}{0ex}{}{[n]}{2}\right)\\ i<j\\ {\sigma }^{-1}{\upsilon }^{-1}\circ \phi \tau (i)>{\sigma }^{-1}{\upsilon }^{-1}\circ \phi \tau (j)\end{array}}{q}_{\dot{\tau (i)},\dot{\tau (j)}}$$

$$=\sum _{\phi \in {\mathfrak{S}}_{\lambda }}\prod _{\begin{array}{c}\{i,j\}\in \left(\genfrac{}{}{0ex}{}{[n]}{2}\right)\\ i<j\\ {\sigma }^{-1}\circ \phi \tau (i)>{\sigma }^{-1}\circ \phi \tau (j)\end{array}}{q}_{\dot{\tau (i)},\dot{\tau (j)}}$$

$$={\mathrm{v}}_n(C_{\dot{\tau }},C_{\sigma }).$$

$${\gamma }_n(C_{\dot{\tau }})$$

$$=\sum _{\dot{\sigma }\in {\mathfrak{S}}_{I_{\lambda }}}\sum _{\upsilon \in {\mathfrak{S}}_{\lambda }}{\mathrm{v}}_n(C_{\dot{\tau }},C_{\upsilon \sigma })C_{\upsilon \sigma }$$

$$=\sum _{\dot{\sigma }\in {\mathfrak{S}}_{I_{\lambda }}}\sum _{\upsilon \in {\mathfrak{S}}_{\lambda }}{\mathrm{v}}_n(C_{\dot{\tau }},C_{\sigma })C_{\upsilon \sigma }$$

$$=\sum _{\dot{\sigma }\in {\mathfrak{S}}_{I_{\lambda }}}{\mathrm{v}}_n(C_{\dot{\tau }},C_{\sigma })\sum _{\upsilon \in {\mathfrak{S}}_{\lambda }}{C}_{\upsilon \sigma }$$

$$=\sum _{\dot{\sigma }\in {\mathfrak{S}}_{I_{\lambda }}}{\mathrm{v}}_n(C_{\dot{\tau }},C_{\sigma })C_{\dot{\sigma }}.$$

$$\mathbf{H}=\Big{\{}\sum_{i=1}^{n}\mu_{i}\mathtt{b}_{i}\ \Big{|}\ n\in\mathbb{N}^{*},\,\mu_{i}\in\mathbb{C}[q_{i,j}],\,\mathtt{b}_{i}\in\mathbf{B}\Big{\}}.$$

$$\displaystyle\mathtt{a}_{i}\,\mathtt{a}_{j_{1}}^{{\dagger}}\dots\mathtt{a}_{j_{t}}^{{\dagger}}=\$$

$$+\sum _{\begin{array}{c}u\in [t]\\ j_u=i\end{array}}{q}_{i,j_1}\dots q_{i,j_{u-1}}{\mathtt{a}}_{j_1}^{†}\dots \widehat{{\mathtt{a}}_{j_u}^{†}}\dots {\mathtt{a}}_{j_t}^{†},$$

$${\mathtt{a}}_i{\mathtt{a}}_{j_1}^{†}\dots {\mathtt{a}}_{j_t}^{†}|0⟩=\sum _{\begin{array}{c}u\in [t]\\ j_u=i\end{array}}{q}_{i,j_1}\dots q_{i,j_{u-1}}{\mathtt{a}}_{j_1}^{†}\dots \widehat{{\mathtt{a}}_{j_u}^{†}}\dots {\mathtt{a}}_{j_t}^{†}|0⟩.$$

$${\mathtt{a}}_{i_s}\dots {\mathtt{a}}_{i_1}{\mathtt{a}}_{j_1}^{†}\dots {\mathtt{a}}_{j_t}^{†}=P{\mathtt{a}}_{i_v}\dots {\mathtt{a}}_{i_1}+Q{\mathtt{a}}_{i_v}\text{with}P,Q\in \mathbf{A}.$$

$${\mathtt{a}}_{i_s}\dots {\mathtt{a}}_{i_1}{\mathtt{a}}_{j_1}^{†}\dots {\mathtt{a}}_{j_t}^{†}={\mathtt{a}}_{j_1}^{†}\dots {\mathtt{a}}_{j_u}^{†}{P}^{\prime }+{\mathtt{a}}_{j_u}^{†}{Q}^{\prime }\text{with}{P}^{\prime },Q^{\prime }\in \mathbf{A}.$$

Full text

Keywords: Quon Algebra, Indefinite Hilbert Module, Hyperplane Arrangement

MSC Number: 05E15, 81R10

1 Introduction

Denote by $\mathbb{C}[q_{i,j}]$ the polynomial ring \mathbb{C}\big{[}q_{i,j}\ \big{|}\ i,j\in\mathbb{N}^{*}\big{]} with variables $q_{i,j}$. The quons are particles whose annihilation and creation operators obey the quon algebra which interpolates between fermions and bosons.

Definition 1.1**.**

By multiparametric quon algebra is meant the free algebra $\mathbf{A}$, equal to \mathbb{C}[q_{i,j}]\big{[}\mathtt{a}_{i}\ \big{|}\ i\in\mathbb{N}^{*}\big{]}, and subject to the anti-involution ${\dagger}$ exchanging $\mathtt{a}_{i}$ with $\mathtt{a}_{i}^{{\dagger}}$ and to the commutation relations

[TABLE]

where $\delta_{i,j}$ is the Kronecker delta.

The multiparametric quon algebra is a generalization of the deformed quon algebra subject to the restriction $q_{i,j}=\bar{q}_{j,i}$ independently studied by Bozejko and Speicher [2, § 3], and by Meljanac and Svrtan [6, § 1.1]. Their algebra is in turn a generalization of the deformed quon algebra investigated by Speicher subject to the restriction $q_{i,j}=q_{j,i}$ [8]. Then, his algebra is a generalization of the quon algebra introduced by Greenberg [5] and studied by Zagier [9, § 1] which is subject to the commutation relations $\mathtt{a}_{i}\mathtt{a}_{j}^{{\dagger}}=q\,\mathtt{a}_{j}^{{\dagger}}\mathtt{a}_{i}+\delta_{i,j}$ obeyed by the annihilation and creation operators of the quon particles, and generating a model of infinite statistics. Finally, the quon algebra is a generalization of the classical Bose and Fermi algebras corresponding to the restrictions $q=1$ and $q=-1$ respectively, as well as of the intermediate case $q=0$ suggested by Hegstrom and investigated by Greenberg [4].

In a Fock-like representation, the generators of $\mathbf{A}$ are the linear operators $\mathtt{a}_{i},\mathtt{a}_{i}^{{\dagger}}:\mathbf{V}\rightarrow\mathbf{V}$ on an infinite dimensional $\mathbb{C}[q_{i,j}]$-vector module $\mathbf{V}$ satisfying the commutation relations

[TABLE]

and the relations

[TABLE]

where $\mathtt{a}_{i}^{{\dagger}}$ is the adjoint of $\mathtt{a}_{i}$, and $|0\rangle$ is a nonzero distinguished vector of $\mathbf{V}$. The $\mathtt{a}_{i}$’s are the annihilation operators and the $\mathtt{a}_{i}^{{\dagger}}$’s the creation operators.

Define the $q_{i,j}$-conjugate $\tilde{P}$ of a monomial $\displaystyle P=\mu\prod_{u\in[n]}q_{i_{u},j_{u}}\in\mathbb{C}[q_{i,j}]$, where $\mu\in\mathbb{C}$, by

[TABLE]

and the $q_{i,j}$-conjugate of a monomial sum $Q=P_{1}+\dots+P_{k}\in\mathbb{C}[q_{i,j}]$ by $\tilde{Q}=\tilde{P}_{1}+\dots+\tilde{P}_{k}$.

Definition 1.2**.**

An indefinite inner product on $\mathbf{V}$ is a map $(.,.):\mathbf{V}\times\mathbf{V}\rightarrow\mathbb{C}[q_{i,j}]$ such that, for $\mu\in\mathbb{C}$, and $\mathtt{u},\mathtt{v},\mathtt{w}\in\mathbf{V}$, we have

• •

$(\mu\mathtt{u},\mathtt{v})=\mu(\mathtt{u},\mathtt{v})$ and $(\mathtt{u}+\mathtt{v},\mathtt{w})=(\mathtt{u},\mathtt{w})+(\mathtt{v},\mathtt{w})$,

• •

$(\mathtt{u},\mathtt{v})=\widetilde{(\mathtt{v},\mathtt{u})}$,

• •

and, if $\mathtt{u}\neq 0$, $(\mathtt{u},\mathtt{u})\in\mathbb{C}[q_{i,j}]\setminus\{0\}$.

Let $\mathbf{H}$ be the submodule of $\mathbf{V}$ generated by the particle states obtained by applying combinations of $\mathtt{a}_{i}$’s and $\mathtt{a}_{i}^{{\dagger}}$’s to $|0\rangle$, that is \mathbf{H}:=\big{\{}\mathtt{a}|0\rangle\ \big{|}\ \mathtt{a}\in\mathbf{A}\big{\}}. The aim of this article is to prove the realizability of that model through the following theorem.

Theorem 1.3**.**

Under the condition $|q_{i,j}|<1$, the module $\mathbf{H}$ is an indefinite Hilbert module for the map $(.,.):\mathbf{H}\times\mathbf{H}\rightarrow\mathbb{C}[q_{i,j}]$ defined, for $\mu,\nu\in\mathbb{C}[q_{i,j}]$, and $\mathtt{a},\mathtt{b}\in\mathbf{A}$, by

[TABLE]

and where the usual bra-ket product $\langle 0|\mathtt{a}\,\mathtt{b}^{{\dagger}}|0\rangle$ is subject to the defining relations of $\mathbf{A}$.

The indefinite inner product of Theorem 1.3 becomes an inner product when the matrix representing $(.,.)$ is positively diagonalizable. Theorem 1.3 is particularly a generalization of the realizability of the deformed quon algebra model for $q_{i,j}=\bar{q}_{j,i}$ established independently by Bozejko and Speicher [2, Corollary 3.2], and by Meljanac and Svrtan [6, Theorem 1.9.4], which in turn is a generalization of the realizability of the deformed quon algebra model for $q_{i,j}=q_{j,i}$ established by Speicher [8, Corollary], which finally is a generalization of the realizability of the quon algebra model established by Zagier [9, Theorem 1].

To prove Theorem 1.3, we first show with Lemma 3.1 that

[TABLE]

is a basis of $\mathbf{H}$, so that we can assume that \displaystyle\mathbf{H}=\Big{\{}\sum_{i=1}^{n}\mu_{i}\mathtt{b}_{i}\ \Big{|}\ n\in\mathbb{N}^{*},\,\mu_{i}\in\mathbb{C}[q_{i,j}],\,\mathtt{b}_{i}\in\mathbf{B}\Big{\}}.

The infinite matrix associated to the map of Theorem 1.3 is \mathbf{M}:=\big{(}(\mathtt{b},\mathtt{a})\big{)}_{\mathtt{a},\mathtt{b}\in\mathbf{B}}.

Let $\displaystyle\left(\!\!{\mathbb{N}^{*}\choose n}\!\!\right)$ be the set of multisets of $n$ elements in $\mathbb{N}^{*}$. We prove with Lemma 3.2 that

[TABLE]

where $\mathfrak{S}_{I}$ is the permutation set of the multiset $I$. For example,

[TABLE]

Proposition 2.1 and Lemma 3.3 permits us to deduce that, if $\displaystyle J\in\binom{\mathbb{N}^{*}}{n}\subseteq\left(\!\!{\mathbb{N}^{*}\choose n}\!\!\right)$, then

[TABLE]

For example, $\det\mathbf{M}_{[3]}=(1-q_{1,2}q_{2,1})^{2}\,(1-q_{1,3}q_{3,1})^{2}\,(1-q_{2,3}q_{3,2})^{2}\,(1-q_{1,2}q_{2,1}q_{1,3}q_{3,1}q_{2,3}q_{3,2})$.

That determinant was independently computed by Meljanac and Svrtan for the specialization $q_{i,j}=\bar{q}_{j,i}$ [6, Theorem 1.9.2], by Duchamp et al. for the specialization $q_{i,j}=q_{j,i}$ [3, § 6.4.1], and by Zagier for the specialization $q_{i,j}=q_{j,i}=q$ [9, Theorem 2].

Moreover, consider the multiset \displaystyle I=\{\overbrace{i_{1},\dots,i_{1}}^{\text{p_{1} times}},\overbrace{i_{2},\dots,i_{2}}^{\text{p_{2} times}},\dots,\overbrace{k,\dots,k}^{\text{p_{k} times}}\}\in\left(\!\!{\mathbb{N}^{*}\choose n}\!\!\right). For $s\in[n]$, let $\dot{s}:=i_{j}$ if $s\in[p_{j}+p_{j-1}+\dots+p_{1}]\setminus[p_{j-1}+\dots+p_{1}]$. Suppose that the generators of $\mathbf{A}$ satisfy the commutation relations $\mathtt{a}_{s}\mathtt{a}_{t}^{{\dagger}}=q_{\dot{s},\dot{t}}\,\mathtt{a}_{t}^{{\dagger}}\mathtt{a}_{s}+\delta_{s,t}$. In that case, if we regard $\mathbf{M}_{[n]}$ as the matrix representing a linear map $\alpha:M\rightarrow M$ on a module $M$, then we prove with Proposition 2.2 and Lemma 3.4 that $\mathbf{M}_{I}$ is the matrix representing $\alpha$ restricted to a submodule $N\subseteq M$ such that $\alpha(N)=N$. Therefore, we can infer that, for every $\displaystyle I\in\left(\!\!{\mathbb{N}^{*}\choose n}\!\!\right)$, $\mathbf{M}_{I}$ is nonsingular for $|q_{i,j}|<1$, $i,j\in\mathbb{N}^{*}$.

When, for special values of the $q_{i,j}$’s, $\mathbf{M}_{[n]}$ is diagonalizable, then $\mathbf{M}_{I}$ becomes positive definite. Indeed, as $\mathbf{M}_{I}$ is the identity matrix if $q_{i,j}=0$, for every $i,j\in\mathbb{N}^{*}$, we deduce by continuity that $\mathbf{M}_{I}$ is positive definite. For these suitable values of $q_{i,j}$, $\mathbf{M}$ becomes a positive definite matrix or, in other terms, the map in Theorem 1.3 becomes an inner product on $\mathbf{H}$. It is the case of the algebras investigated by Meljanac and Svrtan, and Zagier since, with their models, $\mathbf{M}_{[n]}$ is a hermitian matrix, that is consequently diagonalizable.

Finally, we provide another proof of the extended Zagier’s conjecture in Section 4.

Proposition 1.4**.**

Let $n\in\mathbb{N}^{*}$, and assume that the generators of $\mathbf{A}$ satisfy the commutation relations $\mathtt{a}_{s}\mathtt{a}_{t}^{{\dagger}}=q\,\mathtt{a}_{t}^{{\dagger}}\mathtt{a}_{s}+\delta_{s,t}$. Then, each entry of $\mathbf{M}_{[n]}^{-1}$ is an element of \displaystyle\frac{\mathbb{C}[q]}{\displaystyle\prod_{i\in[n-1]}\big{(}1-q^{i^{2}+i}\big{)}^{n-i}}.

The extended Zagier’s conjecture was first established by Meljanac and Svrtan [6, Corollary 2.2.8] who disproved Zagier’s conjecture by using the algorithm of [6, Proposition 2.2.15] for $n=8$. One deduces also Proposition 1.4 from the study of the representation of the permutation group made by Duchamp et al. [3, Proposition 4.6, 4.9].

2 Hyperplane Arrangements

We establish two results we need concerning the hyperplane arrangement associated to the permutation group of $n$ elements in order to prove Theorem 1.3.

Recall that a hyperplane in the space $\mathbb{R}^{n}$ is a $(n-1)$-dimensional linear subspace, and a hyperplane arrangement is a finite set of hyperplanes. For a hyperplane $H$, denote its two associated open half-spaces by $H^{+}$ and $H^{-}$, and let $H^{0}:=H$. A face of a hyperplane arrangement $\mathcal{A}$ is a subset of $\mathbb{R}^{n}$ having the form

[TABLE]

A chamber of $\mathcal{A}$ is a face $C\in F_{\mathcal{A}}$ such that, for every $H\in\mathcal{A}$, $\epsilon_{H}(F)\neq 0$. Denote the set formed by the chambers of $\mathcal{A}$ by $C_{\mathcal{A}}$. Assign a variable $h_{H}^{\varepsilon}$, $\varepsilon\in\{+,-\}$, to every half-space $H^{\varepsilon}$. We work with the polynomial ring R_{\mathcal{A}}:=\mathbb{Z}\big{[}h_{H}^{\varepsilon}\ \big{|}\ H\in\mathcal{A},\,\varepsilon\in\{+,-\}\big{]}, and the module of $R_{\mathcal{A}}$-linear combinations of chambers \displaystyle M_{\mathcal{A}}:=\Big{\{}\sum_{C\in C_{\mathcal{A}}}x_{C}C\ \Big{|}\ x_{C}\in R_{\mathcal{A}}\Big{\}}.

For $C,D\in C_{\mathcal{A}}$, let $H_{C,D}$ be the set of half-spaces \big{\{}H^{\epsilon_{H}(C)}\ \big{|}\ H\in\mathcal{A},\,\epsilon_{H}(C)=-\epsilon_{H}(D)\big{\}}.

The Aguiar-Mahajan bilinear form $\mathrm{v}:M_{\mathcal{A}}\times M_{\mathcal{A}}\rightarrow R_{\mathcal{A}}$ is defined, for $C,D\in C_{\mathcal{A}}$, by

[TABLE]

From $\mathrm{v}$ is defined the linear map $\gamma:M_{\mathcal{A}}\rightarrow M_{\mathcal{A}}$, for $D\in C_{\mathcal{A}}$, by $\displaystyle\gamma(D):=\sum_{C\in C_{\mathcal{A}}}\mathrm{v}(D,C)\,C.$

Let $x=(x_{1},\dots,x_{n})$ be a variable of $\mathbb{R}^{n}$. We precisely investigate the hyperplane arrangement associated to the permutation group $\mathfrak{S}_{n}$ of $n$ elements defined by

[TABLE]

The set formed by the chambers of $\mathcal{A}_{n}$ is

[TABLE]

For $i,j\in[n]$ with $i\neq j$, assign the variable $q_{i,j}$ to the half-space $\{x\in\mathbb{R}^{n}\ |\ x_{i}<x_{j}\}$. On $\mathcal{A}_{n}$, the ring R_{\mathcal{A}_{n}}:=\mathbb{Z}\big{[}q_{i,j}\ \big{|}\ i,j\in[n]\big{]} and the module \displaystyle M_{\mathcal{A}_{n}}:=\Big{\{}\sum_{\sigma\in\mathfrak{S}_{n}}x_{\sigma}C_{\sigma}\ \Big{|}\ x_{\sigma}\in R_{\mathcal{A}_{n}}\Big{\}} are considered. The Aguiar-Mahajan bilinear form becomes $\mathrm{v}_{n}:M_{\mathcal{A}_{n}}\times M_{\mathcal{A}_{n}}\rightarrow R_{\mathcal{A}_{n}}$ defined, for $C_{\sigma},C_{\tau}\in C_{\mathcal{A}_{n}}$, by

[TABLE]

and $\gamma$ the linear map $\gamma_{n}:M_{\mathcal{A}_{n}}\rightarrow M_{\mathcal{A}_{n}}$ defined, for $C_{\tau}\in C_{\mathcal{A}_{n}}$, by

[TABLE]

Proposition 2.1**.**

For an integer $n\geq 2$, we have

[TABLE]

Proof.

We first discuss about the general case of hyperplane arrangements. A flat of $\mathcal{A}$ is an intersection of hyperplanes in $\mathcal{A}$. Denote the set formed by the flats of $\mathcal{A}$ by $L_{\mathcal{A}}$. The weight of a flat $E\in L_{\mathcal{A}}$ is the monomial $\displaystyle\mathrm{b}_{E}:=\prod_{\begin{subarray}{c}H\in\mathcal{A}\\ E\subseteq H\end{subarray}}h_{H}^{+}h_{H}^{-}$, and, if we choose a hyperplane $H$ containing $E$, the multiplicity $\beta_{E}$ of $E$ is half the number of chambers $C\in C_{\mathcal{A}}$ which have the property that $E$ is the minimal edge containing $\overline{C}\cap H$. Aguiar and Mahajan proved that [1, Theorem 8.11]

[TABLE]

Now, concerning $\mathcal{A}_{n}$, let $L_{\mathcal{A}_{n}}^{\prime}=\{E\in L_{\mathcal{A}_{n}}\ |\ \beta_{E}\neq 0\}$. For a subset $I\subseteq[n]$ with $\#I\geq 2$, denote by $E_{I}$ the edge $\displaystyle\bigcap_{\{i,j\}\in\binom{I}{2}}H_{i,j}$. Randriamaro proved that [7, Lemma 3.2, 3.3]

[TABLE]

∎

Take a partition $\lambda=(p_{1},\dots,p_{k})\in\mathrm{Par}(n)$ of $n$. Denote by $\mathfrak{S}_{\lambda}$ the subgroup $\displaystyle\prod_{i\in[k]}\mathfrak{S}_{\lambda_{i}}$ of $\mathfrak{S}_{n}$, where $\mathfrak{S}_{\lambda_{i}}$ is the permutation group of the set $[p_{i}+p_{i-1}+\dots+p_{1}]\setminus[p_{i-1}+\dots+p_{1}]$.

Consider the multiset I_{\lambda}=\{\overbrace{1,\dots,1}^{\text{p_{1} times}},\overbrace{2,\dots,2}^{\text{p_{2} times}},\dots,\overbrace{k,\dots,k}^{\text{p_{k} times}}\}. Denote by $\mathfrak{S}_{I_{\lambda}}$ the permutation set of the multiset $I_{\lambda}$. For $s\in[n]$, define $\dot{s}:=i$ if $s\in[p_{i}+p_{i-1}+\dots+p_{1}]\setminus[p_{i-1}+\dots+p_{1}]$.

Let $\mathrm{p}:\mathfrak{S}_{n}\rightarrow\mathfrak{S}_{I_{\lambda}}$ be the projection $\displaystyle\mathrm{p}(\sigma):=\dot{\sigma(1)}\,\dot{\sigma(2)}\,\dots\,\dot{\sigma(n)}$. For $\dot{\sigma}\in\mathfrak{S}_{I_{\lambda}}$, define the element $\displaystyle C_{\dot{\sigma}}:=\sum_{\sigma\in\mathrm{p}^{-1}(\dot{\sigma})}C_{\sigma}\in M_{\mathcal{A}_{n}}$. Denote by $M_{\mathcal{A}_{n}}^{\lambda}$ the submodule

[TABLE]

For $s,t\in[n]$ with $s\neq t$, assign the variable $q_{\dot{s},\dot{t}}$ to the half-space $\{x\in\mathbb{R}^{n}\ |\ x_{s}<x_{t}\}$.

Proposition 2.2**.**

Let $n\in\mathbb{N}^{*}$, and $\lambda\in\mathrm{Par}(n)$. Then, $\gamma_{n}(M_{\mathcal{A}_{n}}^{\lambda})=M_{\mathcal{A}_{n}}^{\lambda}$.

Proof.

If $\sigma\in\mathfrak{S}_{n}$ such that $\mathrm{p}(\sigma)=\dot{\sigma}\in\mathfrak{S}_{I_{\lambda}}$, then $\mathrm{p}^{-1}(\dot{\sigma})=\mathfrak{S}_{\lambda}\sigma$. Let $\upsilon\sigma\in\mathfrak{S}_{\lambda}\sigma$, and $\dot{\tau}\in\mathfrak{S}_{I_{\lambda}}$. If $\mathrm{p}(\tau)=\dot{\tau}$,

[TABLE]

Hence,

[TABLE]

∎

3 The Bra-Ket Product on $\mathbf{H}$

We prove some useful properties of the map in Theorem 1.3. We particularly connect it to the bilinear form $\mathrm{v}_{n}$ defined in Section 2.

Lemma 3.1**.**

The vector space generated by our particle states is

[TABLE]

Proof.

Let $i\in\mathbb{N}^{*}$. We have,

[TABLE]

where the hat over the $u^{\text{th}}$ term of the product indicates that this term is omitted. So

[TABLE]

Thus, one can recursively remove every annihilation operator $\mathtt{a}_{i}$ of an element $\mathtt{a}|0\rangle\in\mathbf{H}$. ∎

Lemma 3.2**.**

Let $(i_{1},\dots,i_{s})\in(\mathbb{N}^{*})^{s}$ and $(j_{1},\dots,j_{t})\in(\mathbb{N}^{*})^{t}$. If, as multisets, $\{i_{1},\dots,i_{s}\}$ is different from $\{j_{1},\dots,j_{t}\}$, then $\langle 0|\,\mathtt{a}_{i_{s}}\dots\mathtt{a}_{i_{1}}\,\mathtt{a}_{j_{1}}^{{\dagger}}\dots\mathtt{a}_{j_{t}}^{{\dagger}}\,|0\rangle=0$.

Proof.

Suppose that $v$ is the smallest integer in $[s]$ such that $i_{v}\notin\{j_{1},\dots,j_{t}\}\setminus\{i_{1},\dots,i_{v-1}\}.$ Then

[TABLE]

We deduce that $\mathtt{a}_{i_{s}}\dots\mathtt{a}_{i_{1}}\,\mathtt{a}_{j_{1}}^{{\dagger}}\dots\mathtt{a}_{j_{t}}^{{\dagger}}\,|0\rangle=P\,\mathtt{a}_{i_{v}}\dots\mathtt{a}_{i_{1}}\,|0\rangle+Q\,\mathtt{a}_{i_{v}}\,|0\rangle=0$.

Similarly, suppose that $u$ is the smallest integer in $[t]$ such that $j_{u}$ does not belong to the multiset $\{i_{1},\dots,i_{s}\}\setminus\{j_{1},\dots,j_{u-1}\}.$ Then

[TABLE]

And $\langle 0|\,\mathtt{a}_{i_{s}}\dots\mathtt{a}_{i_{1}}\,\mathtt{a}_{j_{1}}^{{\dagger}}\dots\mathtt{a}_{j_{t}}^{{\dagger}}=\langle 0|\,\mathtt{a}_{j_{1}}^{{\dagger}}\dots\mathtt{a}_{j_{u}}^{{\dagger}}\,P^{\prime}+\langle 0|\,\mathtt{a}_{j_{u}}^{{\dagger}}\,Q^{\prime}=0$. ∎

Therefore, we just need to investigate the product $\langle 0|\,\mathtt{a}_{i_{n}}\dots\mathtt{a}_{i_{1}}\,\mathtt{a}_{j_{1}}^{{\dagger}}\dots\mathtt{a}_{j_{n}}^{{\dagger}}\,|0\rangle$ where $(j_{1},\dots,j_{n})$ is a multiset permutation of $(i_{1},\dots,i_{n})$.

Lemma 3.3**.**

Let $\sigma,\tau\in\mathfrak{S}_{n}$, and $C_{\sigma},C_{\tau}\in C_{\mathcal{A}_{n}}$ their associated chambers. Then,

[TABLE]

Proof.

We have

[TABLE]

∎

For $s,t\in[n]$ with $s\neq t$, assign the variable $q_{\dot{s},\dot{t}}$ to the half-space $\{x\in\mathbb{R}^{n}\ |\ x_{s}<x_{t}\}$.

Lemma 3.4**.**

Let $\lambda\in\mathrm{Par(n)}$, and $\dot{\sigma},\dot{\tau}\in\mathfrak{S}_{I_{\lambda}}$. Then, for every $\tau\in\mathrm{p}^{-1}(\dot{\tau})$,

[TABLE]

Proof.

We have

[TABLE]

For every $\sigma\in\mathrm{p}^{-1}(\dot{\sigma})$, and $\tau\in\mathrm{p}^{-1}(\dot{\tau})$, we have, on one side,

[TABLE]

On the other side, if $\mathrm{id}$ is the identity permutation, there exists $\varphi\in\mathfrak{S}_{\lambda}$ such that $\upsilon=\tau^{-1}\varphi\sigma$, and

[TABLE]

Then,

[TABLE]

∎

4 The Conjecture of Zagier

To prove the extended Zagier’s conjecture, we first have to return to the general case of hyperplane arrangements. The set $F_{\mathcal{A}}$ is a monoid with product $FG$ defined, for $F,G\in F_{\mathcal{A}}$, by

[TABLE]

It is also a meet-semilattice with partial order $\preceq$ defined, for $F,G\in F_{\mathcal{A}}$, by

[TABLE]

Denote by $O$ the face in $F_{\mathcal{A}}$ such that, for every $H\in\mathcal{A}$, $\epsilon_{H}(O)=0$. The rank of a face $F\in F_{\mathcal{A}}$ is $\displaystyle\mathrm{rk}\,F:=\dim\bigcap_{\begin{subarray}{c}H\in\mathcal{A}\\ F\subseteq H\end{subarray}}H-\dim O$. The weight of a face $F\in F_{\mathcal{A}}$ is the monomial $\displaystyle\mathrm{b}_{F}:=\prod_{\begin{subarray}{c}H\in\mathcal{A}\\ F\subseteq H\end{subarray}}h_{H}^{+}h_{H}^{-}$. Clearly, $\mathrm{b}_{F}=0$ if $F\in C_{\mathcal{A}}$.

A nested face is a pair $(F,G)$ of faces in $F_{\mathcal{A}}$ such that $F\prec G$. For a nested face $(F,G)$, define the set $F_{\mathcal{A}}^{(F,G)}:=\{K\in F_{\mathcal{A}}\ |\ F\preceq K\preceq G\}$, and the polynomial $\displaystyle\Delta_{F,G}:=\prod_{K\in F_{\mathcal{A}}^{(F,G)}}(1-\mathrm{b}_{K})$.

We prove a variant result of Aguiar and Mahajan [1, Proposition 8.13].

Proposition 4.1**.**

Let $\mathcal{A}$ be a hyperplane arrangement in $\mathbb{R}^{n}$. Each entry of $\gamma^{-1}:M_{\mathcal{A}}\rightarrow M_{\mathcal{A}}$ is an element of $\displaystyle\bigcup_{C\in C_{\mathcal{A}}}\frac{R_{\mathcal{A}}}{\Delta_{O,C}}$.

Proof.

As $\det\gamma$ is a polynomial in $R_{\mathcal{A}}$ with constant term $1$, $\gamma$ is consequently invertible.

For a chamber $D\in C_{\mathcal{A}}$, and a nested face $(A,D)$, let $\displaystyle\mathrm{m}(A,D):=\sum_{\begin{subarray}{c}C\in C_{\mathcal{A}}\\ AC=D\end{subarray}}\mathrm{v}(D,C)\,C$. We prove by backward induction that $\displaystyle\mathrm{m}(A,D)=\sum_{C\in C_{\mathcal{A}}}x_{C}\gamma(C)$ with $\displaystyle x_{C}\in\frac{R_{\mathcal{A}}}{\Delta_{A,D}}$.

We have $\displaystyle\mathrm{m}(D,D)=\gamma(D)$. The opposite of a face $F\in F_{\mathcal{A}}$ is the face $\tilde{F}$ such that, for every $H\in\mathcal{A}$, $\epsilon_{H}(\tilde{F})=-\epsilon_{H}(F)$. One can read in the proof of [1, Proposition 8.13] that

[TABLE]

By induction hypothesis, for every $C\in C_{\mathcal{A}}$, there exists $\displaystyle a_{C}\in\bigcup_{F\in F_{\mathcal{A}}^{(A,D)}\setminus\{A\}}\frac{R_{\mathcal{A}}}{\Delta_{F,D}}$, such that

[TABLE]

Remark that, for every $F\in F_{\mathcal{A}}^{(A,D)}$, we have $\mathrm{b}_{F}=\mathrm{b}_{A\tilde{F}}$, which means that

[TABLE]

Since $A\leq A\tilde{D}$ and $A(\widetilde{A\tilde{D}})=D$, replacing $D$ with $A\tilde{D}$, there exists also $\displaystyle e_{C}\in\bigcup_{F\in F_{\mathcal{A}}^{(A,D)}\setminus\{A\}}\frac{R_{\mathcal{A}}}{\Delta_{F,D}}$, such that, for every $C\in C_{\mathcal{A}}$, $\displaystyle\mathrm{m}(A,A\tilde{D})-(-1)^{\mathrm{rk}\,A\tilde{D}-\mathrm{rk}\,A}\,\mathrm{v}(A\tilde{D},D)\,\mathrm{m}(A,D)=\sum_{C\in C_{\mathcal{A}}}e_{C}\gamma(C)$. Therefore,

[TABLE]

with $\displaystyle\,\frac{a_{C}+(-1)^{\mathrm{rk}\,D-\mathrm{rk}\,A}\,\mathrm{v}(D,A\tilde{D})\,e_{C}}{1-\mathrm{b}_{A}}\,\in\,\bigcup_{F\in F_{\mathcal{A}}^{(A,D)}}\frac{R_{\mathcal{A}}}{\Delta_{F,D}}\,=\,\frac{R_{\mathcal{A}}}{\Delta_{A,D}}$.

For every $D\in C_{\mathcal{A}}$, we have $\mathrm{m}(O,D)=D$. Thus $\displaystyle D=\sum_{C\in C_{\mathcal{A}}}x_{C}\gamma(C)$ with $\displaystyle x_{C}\in\frac{R_{\mathcal{A}}}{\Delta_{O,D}}$, and $\displaystyle\gamma^{-1}(D)=\sum_{C\in C_{\mathcal{A}}}x_{C}\,C$. Finally, each entry of $\gamma^{-1}$ is an element of $\displaystyle\bigcup_{C\in C_{\mathcal{A}}}\frac{R_{\mathcal{A}}}{\Delta_{O,C}}$. ∎

We can deduce the extended Zagier’s conjecture.

Corollary 4.2**.**

Let $n\geq 2$, and suppose that $q_{i,j}=q_{j,i}=q$. Then each entry of $\gamma_{n}^{-1}$ is an element of \displaystyle\frac{\mathbb{Z}[q]}{\displaystyle\prod_{i\in[n-1]}\big{(}1-q^{i^{2}+i}\big{)}^{n-i}}.

Proof.

Let $\displaystyle O_{n}=\bigcap_{H\in\mathcal{A}_{n}}H$, $C_{\sigma}\in C_{\mathcal{A}_{n}}$, and $i,j\in[n-1]$ such that $i\geq j$. A $i$-dimensional face $F_{i}\in F_{\mathcal{A}_{n}}^{(O_{n},C_{\sigma})}$ has the form

[TABLE]

Thus

[TABLE]

If $q=q_{k,l}=q_{l,k}$, then $\mathrm{b}_{F_{i}}=q^{(n-i)^{2}+(n-i)}$. Since $F_{\mathcal{A}_{n}}^{(O_{n},C_{\sigma})}$ contains exactly $i$ $i$-dimensional face, we get \displaystyle\Delta_{O_{n},C_{\sigma}}=\prod_{i\in[n-1]}\big{(}1-q^{(n-i)^{2}+(n-i)}\big{)}^{i}. Finally, we deduce from Proposition 4.1 that each entry of $\gamma_{n}^{-1}$ is an element of \displaystyle\frac{\mathbb{Z}[q]}{\displaystyle\prod_{i\in[n-1]}\big{(}1-q^{(n-i)^{2}+(n-i)}\big{)}^{i}}. ∎