Speaker: Evgeny Smirnov (HSE University, Independent University of Moscow)
Date and time: 30.09.2020, 17:00 (GMT +03:00)
Title: Schubert polynomials for the classical groups
Abstract: A classical result of Borel states that the cohomology ring of the full flag variety GL(n)/B is isomorphic to the polynomial ring in n variables modulo the ideal generated by the symmetric polynomials with zero constant term. On the other hand, this ring has a remarkable basis formed by the Schubert cycles, i.e. the classes of orbit closures of a Borel subgroup in GL(n). In 1970s-80s J. Bernstein, I. Gelfand and S. Gelfand and independently A. Lascoux and M.-P. Schützenberger constructed an explicit collection of representatives of the Schubert cycles, known as the Schubert polynomials. These polynomials have many nice combinatorial properties; they are obtained as the generating functions of certain diagrams (configurations of pseudolines), known as pipe dreams. In particular, their coefficients are positive integers.
The same problem can be considered for generalized flag varieties G/B of other classical groups: G=SO(n) and Sp(2n). The Schubert polynomials for the classical groups of types B/C/D were defined by S. Billey and M. Haiman in 1995; in 2011 T. Ikeda, L. Mihalcea and H. Naruse have studied their T-equivariant analogues, i.e. some “nice” representatives of Schubert classes in the T-equivariant cohomology ring of G/B. I am planning to describe analogues of pipe dreams for these cases, obtained in our recent joint work with Anna Tutubalina.
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