Let p be an odd prime number and let S be a finite set of prime numbers congruent to 1 modulo p. We prove that the group GS(Q)(p) has cohomological dimension 2 if the linking diagram attached to S and p satisfies a certain technical condition, and we show that GS(Q)(p) is a duality group in these cases. Furthermore, we investigate the decomposition behaviour of primes in the extension QS(p)/Q a...

We investigate dual mechanisms for interacting particle systems. Generalizing an approach of Alkemper and Hutzenthaler in the case of coalescing duals, we show that a simple linear transformation leads to a moment duality of suitably rescaled processes. More precisely, we show how dualities of interacting particle systems of the form H(A,B) = q|A∩B|, A,B ⊂ {0, 1} , q ∈ [−1, 1), are rescaled to ...

We introduce the notion of quantum Schur (or q-Schur) superalgebras. These algebras share certain nice properties with q-Schur algebras such as base change property, existence of canonical Z[v, v]-bases, and the duality relation with quantum matrix superalgebra A(m|n). We also construct a cellular Q(υ)-basis and determine its associated cells, called super-cells, in terms of a Robinson–Schenste...

Abstract: Let X be a building of uniform thickness q+1. L-Betti numbers of X are reinterpreted as von-Neumann dimensions of weighted L-cohomology of the underlying Coxeter group. The dimension is measured with the help of the Hecke algebra. The weight depends on the thickness q. The weighted cohomology makes sense for all real positive values of q, and is computed for small q. If the Davis comp...

Let k be a field and Q a finite quiver without oriented cycles. Let kQ be the path algebra of Q and mod kQ the category of k-finite-dimensional right kQ-modules. The cluster category CQ was introduced in [1] (for general Q) and, independently, in [4] (for Q of type An). It is defined as the orbit category of the bounded derived category D(mod kQ) under the action of the automorphism Σ−1 ◦ S, wh...

Abstract We show that quantum Schur–Weyl duality leads to Markov for a variety of asymmetric interacting particle systems. In particular, we consider the following three cases: (1) Using between two-parameter group and Hecke algebra from [6], recover self-duality multi-species ASEP previously discovered in [23] [3]. (2) From co-ideal subalgebra type B [2], find open on semi-infinite line. The f...

Counter-example to (.): If S = {(p,q) : p = q} and r(p,q) = (p,q), one can trivially, for all q, choose p(q) = q to guarantee that r(p(q),q) ∈ S. It is however not possible to find a p which works for all q, indeed the only p which works for a given q is p = q. However, the duality statement (.) holds when S is a half-space {x | v ·x ≥ c}. To see this, define a zero-sum game with scalar p...