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Tarde de Álgebra e Combinatória
Description: Programa:

14h30m - "Homological properties of quantized Schur algebras", Ivan Yudin.

15h15m - "Growth diagrams and non-symmetric 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 Borel-Schur and quantized Schur algebras. In particular, I will explain how projective resolutions of simple modules over quantized Borel-Schur 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 co-Specht modules for Hecke algebras.

This is a joint work with S. Donkin and A. P. Santana.


Title: Growth diagrams and non-symmetric Cauchy identities on near staircases

Speaker: Aram Emami Dashtaki

Abstract: We use Fomin's growth diagrams for Robinson-Schensted-Knuth correspondences to give, on the one hand, a formulation of an analogue of RSK, due to Mason, via reverse RSK, to obtain pairs of semi-skyline augmented fillings, objects which describe combinatorially non-symmetric 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 semi-skyline 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 non-symmetric 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 n-tuples 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).

Place:   Room 2.4 Dmat
Start Date:   2014-03-12
Start Time:   14:30
End Date:   2014-03-12
Research Groups: -Algebra and Combinatorics
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