Two-Rowed Hecke Algebra Representations at Roots of Unity

In this paper, we initiate a study into the explicit construction of irreducible representations of the Hecke algebra $H_n(q)$ of type $A_{n-1}$ in the non-generic case where $q$ is a root of unity. The approach is via the Specht modules of $H_n(q)$ which are irreducible in the generic case, and possess a natural basis indexed by Young tableaux. The general framework in which the irreducible non-generic $H_n(q)$-modules are to be constructed is set up and, in particular, the full set of modules corresponding to two-part partitions is described. Plentiful examples are given.Comment: LaTeX, 9 pages. Submitted for the Proceedings of the 4th International Colloquium ``Quantum Groups and Integrable Systems,'' Prague, 22-24 June 199

A comparison of calculated and measured background noise rates in hard X-ray telescopes at balloon altitude

An actively shielded hard X-ray astronomical telescope has been flown on stratospheric balloons. An attempt is made to compare the measured spectral distribution of the background noise counting rates over the energy loss range 20-300 keV with the contributions estimated from a series of Monte Carlo and other computations. The relative contributions of individual particle interactions are assessed

Hecke algebras of finite type are cellular

Let \cH be the one-parameter Hecke algebra associated to a finite Weyl group $W$, defined over a ground ring in which ``bad'' primes for $W$ are invertible. Using deep properties of the Kazhdan--Lusztig basis of \cH and Lusztig's \ba-function, we show that \cH has a natural cellular structure in the sense of Graham and Lehrer. Thus, we obtain a general theory of ``Specht modules'' for Hecke algebras of finite type. Previously, a general cellular structure was only known to exist in types $A_n$ and $B_n$.Comment: 14 pages; added reference

Cellular structure of qq-Brauer algebras

In this paper we consider the $q$-Brauer algebra over $R$ a commutative noetherian domain. We first construct a new basis for $q$-Brauer algebras, and we then prove that it is a cell basis, and thus these algebras are cellular in the sense of Graham and Lehrer. In particular, they are shown to be an iterated inflation of Hecke algebras of type $A_{n-1}.$ Moreover, when $R$ is a field of arbitrary characteristic, we determine for which parameters the $q$-Brauer algebras are quasi-heredity. So the general theory of cellular algebras and quasi-hereditary algebras applies to $q$-Brauer algebras. As a consequence, we can determine all irreducible representations of $q$-Brauer algebras by linear algebra methods

Homomorphisms and Higher Extensions for Schur algebras and symmetric groups

This paper surveys, and in some cases generalises, many of the recent results on homomorphisms and the higher Ext groups for q-Schur algebras and for the Hecke algebra of type A. We review various results giving isomorphisms between Ext groups in the two categories, and discuss those cases where explicit results have been determined

Representation-theoretic derivation of the Temperley-Lieb-Martin algebras

Explicit expressions for the Temperley-Lieb-Martin algebras, i.e., the quotients of the Hecke algebra that admit only representations corresponding to Young diagrams with a given maximum number of columns (or rows), are obtained, making explicit use of the Hecke algebra representation theory. Similar techniques are used to construct the algebras whose representations do not contain rectangular subdiagrams of a given size.Comment: 12 pages, LaTeX, to appear in J. Phys.

Specht modules and semisimplicity criteria for Brauer and Birman--Murakami--Wenzl Algebras

A construction of bases for cell modules of the Birman--Murakami--Wenzl (or B--M--W) algebra $B_n(q,r)$ by lifting bases for cell modules of $B_{n-1}(q,r)$ is given. By iterating this procedure, we produce cellular bases for B--M--W algebras on which a large abelian subalgebra, generated by elements which generalise the Jucys--Murphy elements from the representation theory of the Iwahori--Hecke algebra of the symmetric group, acts triangularly. The triangular action of this abelian subalgebra is used to provide explicit criteria, in terms of the defining parameters $q$ and $r$, for B--M--W algebras to be semisimple. The aforementioned constructions provide generalisations, to the algebras under consideration here, of certain results from the Specht module theory of the Iwahori--Hecke algebra of the symmetric group

Schur elements for the Ariki-Koike algebra and applications

We study the Schur elements associated to the simple modules of the Ariki-Koike algebra. We first give a cancellation-free formula for them so that their factors can be easily read and programmed. We then study direct applications of this result. We also complete the determination of the canonical basic sets for cyclotomic Hecke algebras of type $G(l,p,n)$ in characteristic 0.Comment: The paper contains the results of arXiv:1101.146

Dual partially harmonic tensors and Brauer-Schur-Weyl duality

Let $V$ be a $2m$-dimensional symplectic vector space over an algebraically closed field $K$. Let \mbb_n^{(f)} be the two-sided ideal of the Brauer algebra \mbb_n(-2m) over $K$ generated by $e_1e_3... e_{2f-1}$, where $0\leq f\leq [n/2]$. Let $\mathcal{HT}_{f}^{\otimes n}$ be the subspace of partially harmonic tensors of valence $f$ in $V^{\otimes n}$. In this paper, we prove that $\dim\mathcal{HT}_f^{\otimes n}$ and \dim\End_{KSp(V)}\Bigl(V^{\otimes n}/V^{\otimes n}\mbb_n^{(f)}\Bigr) are both independent of $K$, and the natural homomorphism from \mbb_n(-2m)/\mbb_n^{(f)} to \End_{KSp(V)}\Bigl(V^{\otimes n}/V^{\otimes n}\mbb_n^{(f)}\Bigr) is always surjective. We show that $\mathcal{HT}_{f}^{\otimes n}$ has a Weyl filtration and is isomorphic to the dual of V^{\otimes n}\mbb_n^{(f)}/V^{\otimes n}\mbb_n^{(f+1)} as a $Sp(V)$-(\mbb_n(-2m)/\mbb_n^{(f+1)})-bimodule. We obtain a $Sp(V)$-\mbb_n-bimodules filtration of $V^{\otimes n}$ such that each successive quotient is isomorphic to some \nabla(\lam)\otimes z_{g,\lam}\mbb_n with \lam\vdash n-2g, \ell(\lam)\leq m and $0\leq g\leq [n/2]$, where \nabla(\lam) is the co-Weyl module associated to \lam and z_{g,\lam} is an explicitly constructed maximal vector of weight \lam. As a byproduct, we show that each right \mbb_n-module z_{g,\lam}\mbb_n is integrally defined and stable under base change

On the idempotents of Hecke algebras

We give a new construction of primitive idempotents of the Hecke algebras associated with the symmetric groups. The idempotents are found as evaluated products of certain rational functions thus providing a new version of the fusion procedure for the Hecke algebras. We show that the normalization factors which occur in the procedure are related to the Ocneanu--Markov trace of the idempotents.Comment: 11 page