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1 Oberwolfach Preprints OWP HERY RANDRIAMARO A Deformed Quon Algebra Mathematisches Forschungsinstitut Oberwolfach ggmbh Oberwolfach Preprints (OWP ISSN

2 Oberwolfach Preprints (OWP Starting in 2007, the MFO publishes a preprint series which mainly contains research results related to a longer stay in Oberwolfach In particular, this concerns the Research in Pairs- Programme (RiP and the Oberwolfach-Leibniz-Fellows (OWLF, but this can also include an Oberwolfach Lecture, for example A preprint can have a size from pages, and the MFO will publish it on its website as well as by hard copy Every RiP group or Oberwolfach-Leibniz-Fellow may receive on request 30 free hard copies (DIN A4, black and white copy by surface mail Of course, the full copy right is left to the authors The MFO only needs the right to publish it on its website wwwmfode as a documentation of the research work done at the MFO, which you are accepting by sending us your file In case of interest, please send a pdf file of your preprint by to rip@mfode or owlf@mfode, respectively The file should be sent to the MFO within 12 months after your stay as RiP or OWLF at the MFO There are no requirements for the format of the preprint, except that the introduction should contain a short appreciation and that the paper size (respectively format should be DIN A4, "letter" or "article" On the front page of the hard copies, which contains the logo of the MFO, title and authors, we shall add a running number (20XX - XX We cordially invite the researchers within the RiP or OWLF programme to make use of this offer and would like to thank you in advance for your cooperation Imprint: Mathematisches Forschungsinstitut Oberwolfach ggmbh (MFO Schwarzwaldstrasse Oberwolfach-Walke Germany Tel Fax URL admin@mfode wwwmfode The Oberwolfach Preprints (OWP, ISSN are published by the MFO Copyright of the content is held by the authors DOI /OWP

3 A Deformed Quon Algebra Hery Randriamaro June 20, 2018 Abstract The quon algebra is an approach to particle statistics in order to provide a theory in which the Pauli exclusion principle and Bose statistics are violated by a small amount The quons are particles whose annihilation and creation operators obey the quon algebra which interpolates between fermions and bosons In this paper we generalize these models by introducing a deformation of the quon algebra generated by a collection of operators a i,k, (i, k N [m], on an infinite dimensional vector space satisfying the deformed q-mutator relations a j,l a i,k qa i,k a j,l + q β k,l δ i,j We prove the realizability of our model by showing that, for suitable values of q, the vector space generated by the particle states obtained by applying combinations of a i,k s and a i,k s to a vacuum state 0 is a Hilbert space The proof particularly needs the investigation of the new statistic cinv and representations of the colored permutation group Keywords: Quon Algebra, Infinite Statistics, Hilbert Space, Colored Permutation Group MSC Number: 05E15, 81R10, 15A15 1 Introduction Let R(q be the fraction field of the real polynomials with variable q By a deformed quon algebra A, we mean the free algebra R(q [ a i,k (i, k N [m] ] subject to the anti-involution exchanging a i,k with a i,k, and to the commutation relation where δ i,j is the Kronecker delta and a j,l a i,k qa i,k a j,l + q β k,l δ i,j, { 0 if l k m mod m β k,l 1 otherwise This algebra is a generalization of the quon algebra introduced by Greenberg [2], subject to the commutation relation a j a i qa i a j + δ i,j obeyed by the annihilation and creation Mathematisches Forschungsinstitut Oberwolfach Schwarzwaldstraße 9-11, Oberwolfach, Germany heryrandriamaro@outlookcom This research was supported through the programme Oberwolfach Leibniz Fellows by the Mathematisches Forschungsinstitut Oberwolfach in

4 operators of the quon particles, and generating a model of infinite statistics Moreover, 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 [1] In a Fock-like representation, the generators of A are the linear operators a i,k, a i,k : V V on an infinite dimensional real vector space V satisfying the commutation relations a j,l a i,k qa i,k a j,l q β k,l δ i,j, and the relations a i,k 0 0, where a i,k is the adjoint of a i,k, and 0 is a nonzero distinguished vector of V The a i,k s are the annihilation operators and the a i,k s the creation operators Let H be the vector subspace of V generated by the particle states obtained by applying combinations of a i,k s and a i,k s to 0, or H : { a 0 a A The aim of this article is to prove the realizability of this model through the following theorem Theorem 11 H is a Hilbert space for the bilinear form (, : H H R(q defined by ( a 0, b 0 : 0 a b 0 with 0 0 1, and for 1 < q < 1 if m 1 and 1 < q < 1 if m > 1 1 m Theorem 11 is a generalization of the realizability of the quon algebra model in infinite statistics proved by Zagier [3, Theorem 1] To prove Theorem 11, we begin by showing in Section 3 that B : { a i n,k n 0 (i u, k u N [m], n N is a basis of H, so that we can assume that { n H λ i b i n N, λ i R(q, b i B i1 Denote by U m the group of all m th roots of unity, and S n the permutation group on [n] We represent an element π of the colored permutation group of m colors U m S n by ( 1 2 n π ( ( (, σ(1, k1 σ(2, k2 σ(n, kn where k 1,, k n [m], and σ is a permutation of [n] But we also adopt the notation π (σ, α meaning that σ S n and α : [n] [m] such that i [n], π(i ( σ(i, α(i 2

5 More generally, let I be a multiset of n elements in N, and S I its permutation set An element θ of the colored permutation set U m S I is defined by θ : (ϕ, ɛ meaning that ϕ S I and ɛ : [n] [m] such that i [n], θ(i ( ϕ(i, ɛ(i Denote the infinite matrix associated to the bilinear form in Theorem 11 by M : ( (f, g f,g B [ ] N Let be the set of multisets of n elements in N n We also prove in Section 3 that M M I with M I ( 0 a ϑ(n a ϑ(1 a θ(1 a θ(n 0 n N I N n ϑ,θ U m S I For m 3 for example, we have 1 q q q q 2 q 2 q q 2 q 2 q q 2 q 2 q 2 q 3 q 3 q 2 q 3 q 3 q 1 q q 2 q q 2 q 2 q q 2 q 2 q q 2 q 3 q 2 q 3 q 3 q 2 q 3 q q 1 q 2 q 2 q q 2 q 2 q q 2 q 2 q q 3 q 3 q 2 q 3 q 3 q 2 q q 2 q 2 1 q q q q 2 q 2 q 2 q 3 q 3 q q 2 q 2 q 2 q 3 q 3 q 2 q q 2 q 1 q q 2 q q 2 q 3 q 2 q 3 q 2 q q 2 q 3 q 2 q 3 q 2 q 2 q q q 1 q 2 q 2 q q 3 q 3 q 2 q 2 q 2 q q 3 q 3 q 2 q q 2 q 2 q q 2 q 2 1 q q q 2 q 3 q 3 q 2 q 3 q 3 q q 2 q 2 q 2 q q 2 q 2 q q 2 q 1 q q 3 q 2 q 3 q 3 q 2 q 3 q 2 q q 2 q M [2] 2 q 2 q q 2 q 2 q q q 1 q 3 q 3 q 2 q 3 q 3 q 2 q 2 q 2 q q q 2 q 2 q 2 q 3 q 3 q 2 q 3 q 3 1 q q q q 2 q 2 q q 2 q 2 q 2 q q 2 q 3 q 2 q 3 q 3 q 2 q 3 q 1 q q 2 q q 2 q 2 q q 2 q 2 q 2 q q 3 q 3 q 2 q 3 q 3 q 2 q q 1 q 2 q 2 q q 2 q 2 q q 2 q 3 q 3 q q 2 q 2 q 2 q 3 q 3 q q 2 q 2 1 q q q q 2 q 2 q 3 q 2 q 3 q 2 q q 2 q 3 q 2 q 3 q 2 q q 2 q 1 q q 2 q q 2 q 3 q 3 q 2 q 2 q 2 q q 3 q 3 q 2 q 2 q 2 q q q 1 q 2 q 2 q q 2 q 3 q 3 q 2 q 3 q 3 q q 2 q 2 q q 2 q 2 q q 2 q 2 1 q q q 3 q 2 q 3 q 3 q 2 q 3 q 2 q q 2 q 2 q q 2 q 2 q q 2 q 1 q q 3 q 3 q 2 q 3 q 3 q 2 q 2 q 2 q q 2 q 2 q q 2 q 2 q q q 1 We need to introduce the statistic cinv : U m S n N defined by cinv (σ, α : #{(i, j [n] 2 i < j, σ(i > σ(j + #{i [n] α(i m Still in Section 3, we prove that M I is the representation of π U m S n q cinv π π on the U m S n module R[U m S I ] Hence if the regular representation of π U m S n q cinv π π, which is M [n], is positive definite, then M I is positive definite We prove in Section 4 that ( (1 n 1 det M [n] + (m 1q (1 q m 1 (1 q i2 +i (n i m n n! (i 2 +i 3 i1

6 We particularly can infer that M [n] is nonsingular for 1 < q < 1 if m 1 and 1 < q < 1 if m > 1 1 m Since M [n] is the identity matrix of order m n n! if q 0, we deduce by continuity that M [n] is positive definite for the values of q mentioned above For these suitable values of q, M is then a symmetric positive definite matrix or, in other terms, the bilinear form of Theorem 11 is an inner product on H But before investigating the deformed quon algebra, it is necessary to recall some notions in representation theory and do some computations in Section 2 We would like to thank Patrick Rabarison for the discussions on quantum statistics 2 Representation Theory We recall the useful notions on representation theory of group and do some calculations for the cyclic groups Take a group G and a finite-dimensional vector space V over a field K Let g, h G, a, b K, and u, v V Then V is a G-module if there is a multiplication of elements of V by elements of G such that u g V (au + bv g a(u g + b(v g, u (gh (u g h, u 1 u where 1 is the neutral element of G Take an element x in the group algebra K[G] Suppose that {v 1,, v n is a basis of V, and that v j x i [n] µ i,jv i Then the representation of x on the G-module V is the matrix R V (x : (µ i,j i,j [n] In particular if x g G λ gg K[G] with λ g R, then the regular representation of x is R K[G] (x : ( λ h 1 g g,h G Lemma 21 Let G be a finite group, H G, and x K[H] Then, det R K[G] (x ( det R K[H] (x G:H Proof Let H {h 1,, h r, and {g 1,, g k be a left coset representative set of H On the ordered basis (g 1 h 1,, g 1 h r, g 2 h 1,, g 2 h r,, g k h 1,, g k h r of K[G], we have where I G:H is the unit matrix of size G : H R K[G] (x R K[H] (x I G:H, Now consider the cyclic group Z m of order m generated by γ, and take a variable z We need the following equalities on the group algebra R(z[Z m ] 4

7 Lemma 22 We have ( det R R(z[Zm] 1 + z γ k ( 1 + (m 1z (1 z m 1 Proof The regular representation of 1 + z γk is the m m circulant matrix with associated polynomial f(x 1 + z j [m 1] xj The determinant of this circulant matrix is i [m] f(ζi If i [m 1], then j [m 1] ζ ij 1 ζi 1 ζ i j [m 1] ζ ij ζi 1 1 ζ i 1 Thus f(1 1 + (m 1z, and f(ζ i 1 z for i [m 1] Lemma 23 We have ( 1 + z γ k 1 1 ( (1 + (m 2z z 1 + (m 1z (1 z Proof The form of 1 + z γk gives us the intuition that its inverse has the form x + y γk The calculation ( 1 + z γ k ( x + y γ k γ k ( x + (m 1zy + zx + ( 1 + (m 2z y confirms the intuition since it leads us to solve the equation system { x + (m 1zy 1 zx + ( 1 + (m 2z y 0 to get the inverse of 1 + z γk We obtain γ k x 1 + (m 2z z ( and y ( 1 + (m 1z (1 z 1 + (m 1z (1 z Lemma 24 We have (1 zγ 1 m z m z i γ i i0 Proof It comes from (1 zγ(1 + zγ + + z m 1 γ m 1 1 z m 5

8 3 The Bilinear Form (, We first show that H is linearly generated by the particle states obtained by applying combinations of a i,k s to 0 Then we prove that M n N M I, where M I is a representation of π U m S n q cinv π π Lemma 31 The vector space generated by our particle states is i1 Proof Let (j, l N [m] We have, I N { n H λ i b i n N, λ i R(q, b i B n a j,l q r a j,l + u [r] i uj q u 1 q β ku,l â i u,k u, where the hat over the u th term of the product indicates that this term is omitted So a j,l 0 u [r] i uj q u 1 q β ku,l â i u,k u 0 Thus one can recursively remove every annihilation operator a j,l of an element a 0 of H Lemma 32 Let ( (j 1, l 1,, (j s, l s (N [m] s and ( (i 1, k 1,, (i r, k r (N [m] r If, as multisets, {j 1,, j s {i 1,, i s, then 0 a js,l s 0 0 Proof Suppose that v is the smallest integer in [s] such that j v / {i 1,, i r \ {j 1,, j v 1 Then a js,l s P a jv,l v + Q a jv,l v with P, Q A We deduce that a js,l s 0 P a jv,l v 0 + Q a jv,l v 0 0 In the same way, suppose that u is the smallest integer in [r] such that i u does not belong to the multiset {j 1,, j s \ {i 1,, i u 1 Then a js,l s a i u,k u P + a i u,k u Q with P, Q A And 0 a js,l s 0 a i u,k u P + 0 a i u,k u Q 0 We just then need to investigate the product 0 a jn,l n a i n,k n 0, where (j 1,, j n is a permutation of (i 1,, i n Consider a multiset I of n elements in N Lemma 33 Let θ, ϑ U m S I Then, 0 a ϑ(n a ϑ(1 a θ(1 a θ(n 0 6 π U m S n ϑθπ q cinv π

9 Proof Let (j 1,, j n be a permutation of (i 1,, i n Then, a jn,l n a i n,k n 0 (u 1,,u n [n] n s [n] i u1 j 1,, i un j n (u 1,,u n [n] n s [n] i u1 j 1,, i un j n (u 1,,u n [n] n i u1 j 1,, i un j n # q σ S n s [n], j si σ(s q us 1 # { r [s 1] u r<u s q β kus, ls 0 q # { r [s 1] u r>u s q β kus, ls 0 q # { (r,s [n] 2 r<s, u r>u s + s [n] β kus, ls 0 { (r,s [n] 2 r<s, σ(r>σ(s + s [n] β k σ(s, ls 0 q cinv π 0 π(σ,α U m S n s [n], j si σ(s, l s k σ(s +α(s mod m We obtain the result by remplacing a jn,l n and a i n,k n by a ϑ(n a ϑ(1 and a θ(1 a θ(n respectively For example, take m 4, ϑ Then ( (2, 4 (5, 1 (2, 4 and θ cinv a 2,4 a 5,1 a 2,4 a 5,2 a 2,3 a 2,1 0 q (2, 1 (1, 3 (3, 3 q 4 + q 5 + q ( (5, 2 (2, 3 (2, 1 cinv (3, 3 (1, 3 (2, 1 Define the multiplication of an element θ (ϕ, ɛ of U m S I by an element π (σ, α of U m S n by θ π (ψ, η U m S I with i [n], ψ(i ϕσ(i, η(i ɛσ(i + α(i mod m Consider the vector space of linear combinations of colored permutations R(q[U m S I ] : { z θ θ z θ R(q θ U m S I One can easily check that, relatively to the multiplication, R(q[U m S I ] is a U m S n module Proposition 34 We have M I R R(q[Um S I ] ( π U m S n q cinv π 7

10 Proof Using Lemma 33, we obtain for θ U m S I θ q cinv π ( π U m S n ϑ U m S I π U m S n ϑθπ q cinv π ϑ 0 a ϑ(n a ϑ(1 a θ(1 a θ(n 0 ϑ ϑ U m S I 4 The Determinant of M [n] We compute the determinant and the inverse of the regular representation of π U m S n q cinv π π Consider the subgroup C n of U m S n defined by C n : { π (σ, α U m S n i [n], σ(i i ( 1 2 i n For i [n], let ξ i be the colored permutation in (1, m (2, m (i, 1 (n, m C n We need the following lemma Lemma 41 We have ( ( (1 det R R(q[Um Sn] q cinv ξ ( m 1 m n n! ξ + (m 1q 1 q ξ C n Proof Remark that ξ C n q cinv ξ ξ ( 1 + q i [n] Then, using Lemma 21 and Lemma 22, we obtain ( det R R(q[Um Sn] 1 + q ξ k ( (1 ( m 1 m n 1 n! i + (m 1q 1 q k [m] ξ k i Now we can compute the determinant of π U m S n q cinv π π Theorem 42 We have ( ( (1 n 1 det R R(q[Um Sn] q cinv π π + (m 1q (1 q m 1 (1 q i2 +i (n i m n n! (i 2 +i π U m S n i1 Proof Every π U m S n has a decomposition π σξ such that σ S n, ξ C n, and cinv π cinv σ + cinv ξ Then, ( ( q cinv π π q cinv σ σ q cinv ξ ξ π U m S n σ S n ξ C n 8

11 It is known that [3, Theorem 2] ( n 1 det R R(q[Sn] q cinv σ σ (1 q i2 +i (n in! σ S n i1 We finally obtain the result by using Lemma 21 and Lemma 41 (i 2 +i For k [n], denote by t k,n the permutation (n n 1 k in cycle notation Let γ n k [n 1] 1 q n k t k,n and ε n k [n] n k i0 q(n k+2i t i k,n 1 q (n k+1(n k+2 Furthermore, let ρ k 1 + (m 2q q i [m 1] ξi k ( 1 + (m 1q (1 q We finish with the inverse of π U m S n q cinv π π Proposition 43 We have Proof We obtain ( 1 q cinv π π ρ i π U m S n i [n] i [n 1] γ i+1 ε i ( ξ Cn qcinv ξ ξ 1 i [n] ρ i by means of Lemma 23 Then [3, Proposition 2] and Lemma 24 permit us to write ( σ S n qcinv σ σ 1 i [n 1] γ i+1ε i References [1] O Greenberg, Example of Infinite Statistics, Physical Review Letters 64 (1990 [2] O Greenberg, Particles with small Violations of Fermi or Bose Statistics, Physical Review D 43 (1991 [3] D Zagier, Realizability of a Model in Infinite Statistics, Communications in Mathematical Physics 147 (

Oberwolfach Preprints

Oberwolfach Preprints OWP 202-07 LÁSZLÓ GYÖFI, HAO WALK Strongly Consistent Density Estimation of egression esidual Mathematisches Forschungsinstitut Oberwolfach ggmbh Oberwolfach Preprints (OWP) ISSN