Langlands duality for representations of quantum groups

We establish a correspondence (or duality) between the characters and the crystal bases of finite-dimensional representations of quantum groups associated to Langlands dual semi-simple Lie algebras. This duality may also be stated purely in terms of semi-simple Lie algebras. To explain this duality, we introduce an "interpolating quantum group" depending on two parameters which interpolates between a quantum group and its Langlands dual. We construct examples of its representations, depending on two parameters, which interpolate between representations of two Langlands dual quantum groups.Comment: 37 pages. References added. Accepted for publication in Mathematische Annale

Polynomial identities for matrices over the Grassmann algebra

We determine minimal Cayley--Hamilton and Capelli identities for matrices over a Grassmann algebra of finite rank. For minimal standard identities, we give lower and upper bounds on the degree. These results improve on upper bounds given by L.\ M\'arki, J.\ Meyer, J.\ Szigeti, and L.\ van Wyk in a recent paper.Comment: 9 page

Remarks on the α\alpha--permanent

We recall Vere-Jones's definition of the $\alpha$--permanent and describe the connection between the (1/2)--permanent and the hafnian. We establish expansion formulae for the $\alpha$--permanent in terms of partitions of the index set, and we use these to prove Lieb-type inequalities for the $\pm\alpha$--permanent of a positive semi-definite Hermitian $n\times n$ matrix and the $\alpha/2$--permanent of a positive semi-definite real symmetric $n\times n$ matrix if $\alpha$ is a nonnegative integer or $\alpha\ge n-1$. We are unable to settle Shirai's nonnegativity conjecture for $\alpha$--permanents when $\alpha\ge 1$, but we verify it up to the $5\times 5$ case, in addition to recovering and refining some of Shirai's partial results by purely combinatorial proofs.Comment: 9 page

Comments on the Deformed W_N Algebra

We obtain an explicit expression for the defining relation of the deformed W_N algebra, DWA(^sl_N)_{q,t}. Using this expression we can show that, in the q-->1 limit, DWA(^sl_N)_{q,t} with t=e^{-2\pi i/N}q^{(k+N)/N} reduces to the sl_N-version of the Lepowsky-Wilson's Z-algebra of level k, ZA(^sl_N)_k. In other words DWA(^sl_N)_{q,t} with t=e^{-2\pi i/N}q^{(k+N)/N} can be considered as a q-deformation of ZA(^sl_N)_k. In the appendix given by H.Awata, S.Odake and J.Shiraishi, we present an interesting relation between DWA(^sl_N)_{q,t} and \zeta-function regularization.Comment: 10 pages, LaTeX2e with ws-ijmpb.cls, Talk at the APCTP-Nankai Joint Symposium on ``Lattice Statistics and Mathematical Physics'', 8-10 October 2001, Tianjin Chin