Programa:
14h30m  "Homological properties of quantized Schur algebras", Ivan Yudin.
15h15m  "Growth diagrams and nonsymmetric Cauchy identities on near staircases", Aram Emami Dashtaki.
16h00m  Coffee break
16h30m  "Inverse Eigenvalue Problems for trees", António Leal Duarte.
****************************************************************** Titles and abstracts ******************************************************************
Title: Homological properties of quantized Schur algebras
Speaker: Ivan Yudin Abstract: The quantized Schur algebras are deformations of the classical Schur algebras. Their representation theory is connected with the representation theory of the Hecke algebras in the same way as the representation theory of the Schur algebras is related to the representation theory of the symmetric groups. In this talk I will focus on the interplay between the categories of modules over quantized BorelSchur and quantized Schur algebras. In particular, I will explain how projective resolutions of simple modules over quantized BorelSchur algebras can be used to construct projective resolutions of Weyl modules over Schur algebras. As a byproduct, we obtain the exactness of the complexes recently constructed by Boltje and Maisch, giving resolutions of the coSpecht modules for Hecke algebras. This is a joint work with S. Donkin and A. P. Santana. **************************************************************** Title: Growth diagrams and nonsymmetric Cauchy identities on near staircases Speaker: Aram Emami Dashtaki Abstract: We use Fomin's growth diagrams for RobinsonSchenstedKnuth correspondences to give, on the one hand, a formulation of an analogue of RSK, due to Mason, via reverse RSK, to obtain pairs of semiskyline augmented fillings, objects which describe combinatorially nonsymmetric Macdonald polynomials; and, on the other hand, an interpretation of the action of crystal operators on biwords whose biletters are cells in a Ferrers shape. We then use these results to characterize, in terms of the shapes of semiskyline augmented fillings, the biwords whose biletters are cells in a near staircase of size n, that is, a staircase of size n, in French convention, plus at most n boxes sited on the stairs, with at most one in each stair. This characterization sheds light on the nonsymmetric Cauchy kernel expansion, restricted to near staircases, due to Lascoux, on the basis of Demazure characters and the basis of Demazure atoms, under the action of appropriate Demazure operators, one for each box sited on the stairs of the staircase. ************************************************************************ Title: Inverse Eigenvalue Problems for trees Speaker: António Leal Duarte. Abstract: The Inverse Eigenvalue Problem IEP for a tree T (with vertices 1, ... , n) consists in describing the set of all ntuples of real numbers that may occur as eigenvalues of real symmetric matrices A with graph T (that is a non diagonal element of A in position (i, j) is nonzero if and only if there is an edge between i and j in T); the set of nxn symmetric matrices with graph T is denoted by S(T). It is known that any distinct n real numbers may occur as eigenvalues of ne of those matrices, but the case of multiple eigenvalues is far from being knwon. This seems to be a very difficult problem and highly combinatorial. Even the apparently simpler problem of just describing the possible lists of multiplicities that may occur among the eigenvalues of matrices in S(T) seems very difficult, depending heavily on the graph. Some related questions we will discuss are: (i) What is the maximum possible multiplicities of eigenvalues for matrices in S(G); (ii) what are the minimum number of distinct eigenvalues for A in S(G); (iii) what is the minimum number of multiplicity one eigenvalues for A in S(T).
