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Let $S_n$ denote the symmetric group on $\{1,2,\ldots,n\}$. For two permutations $u, v\in S_n$ such that $u\leq v$ in the Bruhat order, let $R_{u,v}(q)$ and $\R_{u,v}(q)$ denote the Kazhdan-Lusztig $R$-polynomial and $\R$-polynomial,…

Let $R$ be a commutative ring with identity and $n\geq1$ be an integer. Let $R^{n}=R\times\cdots\times R~(n~times)$. The \textit{total dot product} graph, denoted by $TD(R,n)$ is a simple graph with elements of $R^{n}-\{(0,0,\ldots,0)\}$ as…

Singular nonsymmetric Macdonald polynomials are constructed by use of the representation theory of the Hecke algebras of the symmetric groups. These polynomials are labeled by quasistaircase partitions and are associated to special…

We consider certain modules of the symmetric groups whose basis elements are called tabloids. Some of these modules are isomorphic to subspaces of the cohomology rings of subvarieties of flag varieties as modules of the symmetric groups. We…

Let R be a ring. Let SSE-R be the equivalence relation on square matrices (allowed to have different size) over R generated by A ~ B if there exist matrices U,V over R such that A = UV and B = VU . An invariant of SSE-R is shift equivalence…

Let $Y$ be a scheme in which 2 is invertible and let $V$ be a rank $n$ vector bundle on $Y$ endowed with a non-degenerate symmetric bilinear form $q$. The orthogonal group ${\bf O}(q)$ of the form $q$ is a group scheme over $Y$ whose…

Let $\hat{A}_n$ be the completion by the degree of a differential operator of the $n$-th Weyl algebra with generators $x_1,\ldots,x_n,\partial^1,\ldots,\partial^n$. Consider $n$ elements $X_1,\ldots,X_n$ in $\hat{A}_n$ of the form $$ X_i =…

The classical Noether Normalization Lemma states that if $S$ is a finitely generated algebra over a field $k$, then there exist elements $x_1,\dots,x_n$ which are algebraically independent over $k$ such that $S$ is a finite module over…

The goal of this paper is to lay the foundations for a combinatorial study, via orthogonal functions and intertwining operators, of category O for the rational Cherednik algebra of type G(r,p,n). As a first application, we give a…

Let $R$ be a positively graded algebra over a field. We say that $R$ is Hilbert-cyclotomic if the numerator of its reduced Hilbert series has all of its roots on the unit circle. Such rings arise naturally in commutative algebra, numerical…

Consider S_n, the symmetric group on n letters, and let maj pi denote the major index of a permutation pi in S_n. Given positive integers k,l and nonnegative integers i,j, define m_n^{k,l}(i,j) := number of pi in S_n such that maj pi = i…

We introduce controlled $KK$-theory groups associated to a pair $(A,B)$ of separable $C^*$-algebras. Roughly, these consist of elements of the usual $K$-theory group $K_0(B)$ that approximately commute with elements of $A$. Our main results…

We introduce the multiset partition algebra, ${\rm M\!P}_{r,k}(x)$, that has bases elements indexed by multiset partitions, where $x$ is an indeterminate and $r$ and $k$ are non-negative integers. This algebra can be realized as a diagram…

To a finite quiver equipped with a positive integer on each of its vertices, we associate a holomorphic symplectic manifold having some parameters. This coincides with Nakajima's quiver variety with no stability parameter/framing if the…

We define a quantity $c_m(n,k)$ as a generalization of the notion of the composition of the positive integer $n$ into $k$ parts. We proceed to derive some known properties of this quantity. In particular, we relate two partial Bell…

We develop a theory of R-module Thom spectra for a commutative symmetric ring spectrum R and we analyze their multiplicative properties. As an interesting source of examples, we show that R-algebra Thom spectra associated to the special…

We construct a quotient ring of the ring of diagonal coinvariants of the complex reflection group $W=G(m,p,n)$ and determine its graded character. This generalises a result of Gordon for Coxeter groups. The proof uses a study of category…

In this paper we define a two-variable, generic Hecke algebra, H, for each complex reflection group G(b,1,n). The algebra H specializes to the group algebra of G(b,1,n) and also to an endomorphism algebra of a representation of GL(n,q)…

We construct modular categories from Hecke algebras at roots of unity. For a special choice of the framing parameter, we recover the Reshetikhin-Turaev invariants of closed 3-manifolds constructed from the quantum groups U_q sl(N) by…

We consider twisted standard filtrations of Soergel bimodules associated to arbitrary Coxeter groups and show that the graded multiplicities in these filtrations can be interpreted as structure constants in the Hecke algebra. This…