A compactly generated group whose Hecke algebras admit no bounds on their representations

We construct a compactly generated, totally disconnected, locally compact group whose Hecke algebra with respect to any compact open subgroup does not have a C∗-enveloping algebra. 2000 Mathematics Subject Classification. 20C08

On the Existence of Configurations of Subspaces in a Hilbert Space with Fixed Angles

For a class of $*$-algebras, where $*$-algebra $A_{\Gamma,\tau}$ is generated by projections associated with vertices of graph $\Gamma$ and depends on a parameter $\tau$ $(0 < \tau \leq 1)$, we study the sets $\Sigma_\Gamma$ of values of $\tau$ such that the algebras $A_{\Gamma,\tau}$ have nontrivial $*$-representations, by using the theory of spectra of graphs. In other words, we study such values of $\tau$ that the corresponding configurations of subspaces in a Hilbert space exist.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Multiplicative Bases for the Centres of the Group Algebra and Iwahori-Hecke Algebra of the Symmetric Group

Let \H_n be the Iwahori-Hecke algebra of the symmetric group $S_n$, and let Z(\H_n) denote its centre. Let $B={b_1,b_2,...,b_t}$ be a basis for Z(\H_n) over $R=\Z[q,q^{-1}]$. Then $B$ is called \emph{multiplicative} if, for every $i$ and $j$, there exists $k$ such that $b_ib_j= b_k$. In this article we prove that there are no multiplicative bases for $Z(\Z S_n)$ and Z(\H_n) when $n\ge 3$. In addition, we prove that there exist exactly two multiplicative bases for $Z(\Z S_2)$ and none for Z(\H_2).Comment: 6 pages. To appear in Proceedings of the Southeastern Lie Theory Workshop Series, Proceedings of Symposia in Pure Mathematic

Power sums and Homfly skein theory

The Murphy operators in the Hecke algebra H_n of type A are explicit commuting elements, whose symmetric functions are central in H_n. In [Skein theory and the Murphy operators, J. Knot Theory Ramif. 11 (2002), 475-492] I defined geometrically a homomorphism from the Homfly skein C of the annulus to the centre of each algebra H_n, and found an element P_m in C, independent of n, whose image, up to an explicit linear combination with the identity of H_n, is the m-th power sum of the Murphy operators. The aim of this paper is to give simple geometric representatives for the elements P_m, and to discuss their role in a similar construction for central elements of an extended family of algebras H_{n,p}.Comment: Published by Geometry and Topology Monographs at http://www.maths.warwick.ac.uk/gt/GTMon4/paper15.abs.htm

On the irreducible Specht modules for Iwahori--Hecke algebras of type A with q=−1q=-1

Let $p$ be a prime and $\mathbb{F}$ a field of characteristic $p$, and let $\mathcal{H}_n$ denote the Iwahori--Hecke algebra of the symmetric group $\mathfrak{S}_n$ over $\mathbb{F}$ at $q=-1$. We prove that there are only finitely many partitions $\lambda$ such that both $\lambda$ and $\lambda'$ are 2-singular and the Specht module $S^\lambda$ for \mathcal{H}_{|\la|} is irreducible

Center of Twisted Graded Hecke Algebras for Homocyclic Groups

We determine explicitly the center of the twisted graded Hecke algebras associated to homocyclic groups. Our results are a generalization of formulas by M. Douglas and B. Fiol in [J. High Energy Phys. 2005 (2005), no. 9, 053, 22 pages, hep-th/9903031]

On graded decomposition numbers for cyclotomic Hecke algebras in quantum characteristic 2

Brundan and Kleshchev introduced graded decomposition numbers for representations of cyclotomic Hecke algebras of type $A$, which include group algebras of symmetric groups. Graded decomposition numbers are certain Laurent polynomials, whose values at 1 are the usual decomposition numbers. We show that in quantum characteristic 2 every such polynomial has non-zero coefficients either only in odd or only in even degrees. As a consequence, we find the first examples of graded decomposition numbers of symmetric groups with non-zero coefficients in some negative degrees.Comment: 6 pages. Definition of parity function $\epsilon$ corrected for $l>1$. Comments welcome. To appear in Bulletin of the London Mathematical Societ

Row and column removal in the q-deformed Fock space

Analogues of James's row and column removal theorems are proved for the q-decomposition numbers arising from the canonical basis in the q-deformed Fock space

On the representation dimension of skew group algebras, wreath products and blocks of Hecke algebras

We establish bounds for the representation dimension of skew group algebras and wreath products. Using this, we obtain bounds for the representation dimension of a block of a Hecke algebra of type A, in terms of the weight of the block. This includes certain blocks of group algebras of symmetric groups.Comment: 8 page