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13
qGaussian processes: Noncommutative and classical aspects
 Commun. Math. Phys
, 1997
"... Abstract. We examine, for −1 < q < 1, qGaussian processes, i.e. families of operators (noncommutative random variables) Xt = at + a ∗ t – where the at fulfill the qcommutation relations asa ∗ t − qa ∗ t as = c(s, t) · 1 for some covariance function c(·, ·) – equipped with the vacuum expectation ..."
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Cited by 64 (2 self)
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Abstract. We examine, for −1 < q < 1, qGaussian processes, i.e. families of operators (noncommutative random variables) Xt = at + a ∗ t – where the at fulfill the qcommutation relations asa ∗ t − qa ∗ t as = c(s, t) · 1 for some covariance function c(·, ·) – equipped with the vacuum expectation state. We show that there is a qanalogue of the Gaussian functor of second quantization behind these processes and that this structure can be used to translate questions on qGaussian processes into corresponding (and much simpler) questions in the underlying Hilbert space. In particular, we use this idea to show that a large class of qGaussian processes possess a noncommutative kind of Markov property, which ensures that there exist classical versions of these noncommutative processes. This answers an old question of Frisch and Bourret [FB].
Advanced Determinant Calculus
, 1999
"... The purpose of this article is threefold. First, it provides the reader with a few useful and efficient tools which should enable her/him to evaluate nontrivial determinants for the case such a determinant should appear in her/his research. Second, it lists a number of such determinants that have ..."
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Cited by 37 (0 self)
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The purpose of this article is threefold. First, it provides the reader with a few useful and efficient tools which should enable her/him to evaluate nontrivial determinants for the case such a determinant should appear in her/his research. Second, it lists a number of such determinants that have been already evaluated, together with explanations which tell in which contexts they have appeared. Third, it points out references where further such determinant evaluations can be found.
Noncommutative Symmetric Functions III: Deformations Of Cauchy And Convolution Algebras
"... This paper discusses various deformations of free associative algebras and of their convolution algebras. Our main examples are deformations of noncommutative symmetric functions related to families of idempotents in descent algebras, and a simple qanalogue of the shuffle product, which has unexpec ..."
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Cited by 22 (8 self)
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This paper discusses various deformations of free associative algebras and of their convolution algebras. Our main examples are deformations of noncommutative symmetric functions related to families of idempotents in descent algebras, and a simple qanalogue of the shuffle product, which has unexpected connections with quantum groups, hyperplane arrangements, and certain questions in theoretical physics (the quon algebra).
A qDeformation of a Trivial Symmetric Group Action
"... Let A be the arrangement of hyperplanes consisting of the reflecting hyperplanes for the root system A n\Gamma1 . Let B = B(q) be the Varchenko matrix for this arrangement with all hyperplane parameters equal to q. We show that B is the matrix with rows and columns indexed by permutations with oe; ..."
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Cited by 6 (2 self)
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Let A be the arrangement of hyperplanes consisting of the reflecting hyperplanes for the root system A n\Gamma1 . Let B = B(q) be the Varchenko matrix for this arrangement with all hyperplane parameters equal to q. We show that B is the matrix with rows and columns indexed by permutations with oe; entry equal to q i(oe where i() is the number of inversions of . Equivalently B is the matrix for left multiplication on C S n by 2Sn i() Clearly B commutes with the rightregular action of S n on C S n . A general theorem of Varchenko applied in this special case shows that B is singular exactly when q is a j(j \Gamma 1) root of 1 for some j between 2 and n. In this paper we prove two results which partially solve the problem (originally posed by Varchenko) of describing the S n module structure of the nullspace of B in the case that B is singular. Our first result is that ker(B) = ind in the case that q = e where Lie n denotes the multilinear part of the free Lie algebra with n generators. Our second result gives an elegant formula for the determinant of B restricted to the virtual S n module P with characteristic the power sum symmetric function p j (x). Section 1:
Study of Gram matrices in Fock representation of multiparametric canonical commutation relations, extended Zagier’s conjecture, hyperplane arrangements and quantum groups
 Math. Commun
, 1996
"... Abstract.In this Colloqium Lecture (by one of the authors (D.S)) a thorough presentation of the authors research on the subjects,stated in the title,is given.By quite laborious mathematics it is explained how one can handle systems in which each Heisenberg commutation relation is deformed separately ..."
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Cited by 5 (1 self)
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Abstract.In this Colloqium Lecture (by one of the authors (D.S)) a thorough presentation of the authors research on the subjects,stated in the title,is given.By quite laborious mathematics it is explained how one can handle systems in which each Heisenberg commutation relation is deformed separately.For Hilbert space realizability a detailed determinant computations (extending Zagier’s one parametric formulas) are carried out.The inversion problem of the associated Gram matrices on Fock weight spaces is completely solved (Extended Zagier’s conjecture) and a counterexample to the original Zagier’s conjecture is presented in detail. Saˇzetak.U ovom Kolokviju (jednog od autora (D.S)) cjelovito su prikazana istraˇzivanja autora o temama formuliranima u naslovu.S poprilično matematike objaˇsnjeno je kako se mogu obradivati sustavi u kojima je svaka Heisenbergova komutacijska relacija deformirana odvojeno.Za realizabilnost na Hilbertovu prostoru provedeno je detaljno računanje determinanata (koje proˇsiruje Zagierove jednoparametarske formule).Problem inverzije pridruˇzenih Gramovih matrica na Fockovim teˇzinskim prostorima je potpuno rijeˇsen (Proˇsirena Zagierova hipoteza) i kontraprimjer za originalnu Zagierovu hipotezu je detaljno prikazan. Key words and phrases.Multiparametric canonical commutation relations,deformed partial derivatives,lattice of subdivisions,deformed regular representation,quantum bilinear form,Zagier’s conjecture. Ključne riječi i pojmovi.Multiparametarske kanonske komutacijske relacije,deformirane parcijalne derivacije,reˇsetka subdivizija,deformirana regularna reprezentacija, kvantna bilinearna forma,Zagierova hipoteza.
On The Smith Normal Form Of The Varchenko Bilinear Form Of A Hyperplane Arrangement
, 1997
"... this paper we will study the Smith Normal Form of the matrices B. ..."
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Cited by 2 (1 self)
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this paper we will study the Smith Normal Form of the matrices B.
On the Action of a qDeformation of the Trivial Idempotent on the Group Algebra of the Symmetric Group
"... Let A be the arrangement of hyperplanes consisting of the reflecting hyperplanes for the root system A n\Gamma1 . Let B = B(q) be the Varchenko matrix for this arrangement with all hyperplane parameters equal to q. We show that B is the matri x with rows and columns indexed by permutations with oe; ..."
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Let A be the arrangement of hyperplanes consisting of the reflecting hyperplanes for the root system A n\Gamma1 . Let B = B(q) be the Varchenko matrix for this arrangement with all hyperplane parameters equal to q. We show that B is the matri x with rows and columns indexed by permutations with oe; ø entry equal to q i(oeø \Gamma1 ) where i(ß) is the number of inversions of ß. Equivalently B is the matrix for left multiplication on CS n by \Gamma n (q) = X ß2Sn q i(ß) ß: Clearly B commutes with the rightregular action of S n on CS n . A general theorem of Varchenko applied in this special case shows that B is singular exactly when q is a j(j \Gamma 1) st root of 1 for some j between 2 and n. In this paper we prove two results which partially solve the problem (originally posed by Varchenko) of describing the S n module structure of the nullspace of B in the case that B is singular. Our first result is that ker(B) = ind Sn Sn\Gamma1 (Lie n\Gamma1 )=Lie n in the case tha...
Some ALgebraic Properties . . .
, 1999
"... We examine a bilinear form associated with a real arrangement of hyperplanes introduced in [Schechtman and Varchenko 1991]. Our main objective is to show that the linear algebraic properties of this bilinear form are related to the combinatorics and topology of the hyperplane arrangement. We will su ..."
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We examine a bilinear form associated with a real arrangement of hyperplanes introduced in [Schechtman and Varchenko 1991]. Our main objective is to show that the linear algebraic properties of this bilinear form are related to the combinatorics and topology of the hyperplane arrangement. We will survey results and state a number of open problems which relate the determinant, cokernel structure and Smith normal form of the bilinear form to combinatorial and topological invariants of the arrangement including the characteristic polynomial, combinatorial structure of the intersection lattice and homology of the Milnor fibre.