Related papers: A q-analogue of an identity of N. Wallach

We construct a subalgebra of the Hecke algebra of type A. This is a generalization of the group algebra of the alternating groups. All the equivalent classes of irreducible representations of the subalgebra and the q-analogue of the…

We give a $q$-analogue of some binomial coefficient identities of Y. Sun [Electron. J. Combin. 17 (2010), #N20] as follows: {align*} \sum_{k=0}^{\lfloor n/2\rfloor}{m+k\brack k}_{q^2}{m+1\brack n-2k}_{q} q^{n-2k\choose 2} &={m+n\brack…

Let $\bm{i}=1+q+...+q^{i-1}$. For certain sequences $(r_1,...,r_l)$ of positive integers, we show that in the Hecke algebra $\mathscr{H}_n(q)$ of the symmetric group $\mathfrak{S}_n$, the product $(1+\bm{r_1}T_{r_1})... (1+\bm{r_l}T_{r_l})$…

We introduce an algebra which describes the multiplication structure of a family of q-series containing a q-analogue of multiple zeta values. The double shuffle relations are formulated in our framework. They contain a q-analogue of…

We find a $q$-analog of the following symmetrical identity involving binomial coefficients $\binom{n}{m}$ and Eulerian numbers $A_{n,m}$, due to Chung, Graham and Knuth [{\it J. Comb.}, {\bf 1} (2010), 29--38]: {equation*} \sum_{k\geq…

The 0-Hecke algebra $H_n(0)$ is a deformation of the group algebra of the symmetric group $\SS_n$. We show that its coinvariant algebra naturally carries the regular representation of $H_n(0)$, giving an analogue of the well-known result…

Let $H=H_q(n)$ be the Hecke algebra of the symmetric group of degree n, over a field of arbitrary characteristic, and where q is a primitive l-th root of unity in $K$. Let $H_{\rho}$ be an l-parabolic subalgebra of $H$. We give an…

We study a family of shuffling operators on the symmetric group $S_n$, which includes the top-to-random shuffle. The general shuffling scheme consists of removing one card at a time from the deck (according to some probability distribution)…

It is shown that a refined version of a q-analogue of the Eulerian numbers together with the action, by conjugation, of the subgroup of the symmetric group $S_n$ generated by the $n$-cycle $(1,2,...,n)$ on the set of permutations of fixed…

We study the algebra $\Sigma_n$ induced by the action of the symmetric group $S_n$ on $V^{\otimes n}$ when $\dim V=2$. Our main result is that the space of symmetric elements of $\Sigma_n$ is linearly spanned by the involutions of $S_n$.

Let $\mathfrak S_{[i,j]}$ be the subgroup of the symmetric group $\mathfrak S_n$ generated by adjacent transpositions $(i,i+1), \dotsc, (j-1,j)$, assuming $1 \leq i < j \leq n$. We give a combinatorial rule for evaluating induced sign…

Analogues of 1-shuffle elements for complex reflection groups of type $G(m,1,n)$ are introduced. A geometric interpretation for $G(m,1,n)$ in terms of rotational permutations of polygonal cards is given. We compute the eigenvalues, and…

The fundumental invariant of the Hecke algebra $H_{n}(q)$ is the $q$-deformed class-sum of transpositions of the symmetric group $S_{n}$. Irreducible representations of $H_{n}(q)$, for generic $q$, are shown to be completely characterized…

Inside the double affine Hecke algebra of type $GL_n$, which depends on two parameters $q$ and $\tau$, we define a subalgebra $\mathbb{H}^{\mathfrak{gl}_n}$ that may be thought of as a $q$-analogue of the degree zero part of the…

The Iwahori-Hecke algebra of the symmetric group is the convolution algebra of $\gl_n$-invariant functions on the variety of pairs of complete flags over a finite field. Considering convolution on the space of triples of two flags and a…

We describe an efficient algorithm to write any element of the alternating group A_n as a product of two n-cycles (in particular, we show that any element of A_n can be so written -- a result of E. A. Bertram). An easy corollary is that…

Recently N.Jing discovered a certain combinatorial identity from validity of the Serre relations in some vertex representations of quantum Kac-Moody algebras. We generalize this identity, in particular, extending it from polynomials to…

We prove a q-analogue of the formula $ \sum_{1\le k\le n} \binom nk(-1)^{k-1}\sum_{1\le i_1\le i_2\le... \le i_m=k}\frac1{i_1i_2... i_m} = \sum_{1\le k\le n}\frac{1}{k^m} $ by inverting a formula due to Dilcher.

The *somewhere-to-below shuffles* are the elements \[ t_{\ell} := \operatorname{cyc}_{\ell}+\operatorname{cyc}_{\ell,\ell+1}+\operatorname{cyc}_{\ell,\ell+1,\ell+2}+\cdots+\operatorname{cyc}_{\ell,\ell+1,\ldots,n} \] (for $\ell \in…

We study a $q$-deformation for the semi-direct product of the symmetric group $S_n$ with the Clifford algebra on $n$ anticommuting generators. We obtain a $q$-version of the projective analogue for the classical Young symmetrizer found by…