Related papers: Factorizations in Hecke algebras I: long cycle fac…
Recent work by Huang, Lewis, Morales, Reiner, and Stanton suggests that the regular elliptic elements of $\mathrm{GL}_n \mathbb{F}_q$ are somehow analogous to the $n$-cycles of the symmetric group. In 1981, Stanley enumerated the…
We study the von Neumann algebra generated by q--deformed Gaussian elements l_i+l_i^* where operators l_i fulfill the q--deformed canonical commutation relations l_i l_j^*-q l_j^* l_i=delta_{ij} for -1<q<1. We show that if the number of…
Permutation polynomials are of particular significance in several areas of applied mathematics, such as Coding theory and Cryptography. Many recent constructions are based on the Akbary-Ghioca-Wang (AGW) criterion. Along this line of…
Let $R$ be a commutative ring with identity and let $V$ be a free $R$-module of rank $n$ for some $n\in\mathbb{N}$. Fixing an $R$-basis $\mathcal{E}$ of $V$, the symmetric group $\mathfrak{S}_n$ acts on $V$ by permuting $\mathcal{E}$ and…
We define a category of planar diagrams whose Grothendieck group contains an integral version of the infinite rank Heisenberg algebra, thus yielding a categorification of this algebra. Our category, which is a q-deformation of one defined…
A definition is offered of the factorial characters of the general linear group, the symplectic group and the orthogonal group in an odd dimensional space. It is shown that these characters satisfy certain flagged Jacobi-Trudi identities.…
We prove that the mixed $q$-Gaussian algebra $\Gamma_{Q}(H_{\mathbb{R}})$ associated to a real Hilbert space $H_{\mathbb{R}}$ and a real symmetric matrix $Q=(q_{ij})$ with $\sup|q_{ij}|<1$, is a factor as soon as $\dim H_{\mathbb{R}}\geq2$.…
We present algorithms to factorize weighted homogeneous elements in the first polynomial Weyl algebra and $q$-Weyl algebra, which are both viewed as a $\mathbb{Z}$-graded rings. We show, that factorization of homogeneous polynomials can be…
The problem of computing the class expansion of some symmetric functions evaluated in Jucys-Murphy elements appears in different contexts, for instance in the computation of matrix integrals. Recently, M. Lassalle gave a unified algebraic…
We examine two isomorphisms between affine Hecke algebras of type $A$ associated with parameters $q^{-1}$, $t^{-1}$ and $q$, $t$. One of them maps the non-symmetric Macdonald polynomials $E_{\eta}(x;q^{-1},t^{-1})$ onto $E_{\eta}(x;q,t)$,…
In this paper, we look at the problem of determining the composition factors for the affine graded Hecke algebra via the computation of Kazhdan-Lusztig type polynomials. We review the algorithms of \cite{L1,L2}, and use them in particular…
We prove some unique prime factorization results for tensor products of type $II_1$ factors of the form $\Gamma_q(\mathbb{C}, S \otimes H)$ arising from symmetric independent copies with sub-exponential dimensions of the spaces $D_k(S)$ and…
We evaluate combinatorially certain connection coefficients of the symmetric group that count the number of factorizations of a long cycle as a product of three permutations. Such factorizations admit an important topological interpretation…
We construct the scattering matrices for an arbitrary Weyl group in terms of elementary operators which obey the generalised Yang-Baxter equation. We use this construction to obtain the affine Hecke algebras. The center of the affine Hecke…
We introduce a type $A$ crystal structure on decreasing factorizations of fully-commutative elements in the 0-Hecke monoid which we call $\star$-crystal. This crystal is a $K$-theoretic generalization of the crystal on decreasing…
We introduce a new approach to an enumerative problem closely linked with the geometry of branched coverings; that is, we study the number of ways a permutation can be decomposed into a product of a given number of 2-cycles, 3-cycles, etc.…
Our aim in this thesis is to use the language of deformation-quantization to understand certain quantized algebras by looking at properties of the corresponding commutative ones, and conversely to obtain results about the commutative…
We investigate the problem of defining group or loop structures on spheres, where by ''sphere'' we mean the level set q(x) = c of a general K-valued quadratic form q, for an invertible scalar c. When K is a field and q non-degenerate, then…
The aim of this paper is to introduce and study a large class of $\mathfrak{g}$-module algebras which we call factorizable by generalizing the Gauss factorization of (square or rectangular) matrices. This class includes coordinate algebras…
We extend the deformation theory algorithm of matrix factorizations to systems with more than one D-brane. The obstructions to the deformations are F-term equations which can be integrated to an effective superpotential. We demonstrate the…