Combinatorial aspects of geometry and representation theory

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

Abstract for award of Haiman DMS-0301072 Professor Haiman works on problems in combinatorics related to algebra, geometry and representation theory. Using new discoveries on the geometry of Hilbert schemes, he has recently proved the positivity conjecture for Macdonald polynomials and established a formula in terms of these polynomials for the character of a "doubled" version of the classical spaces of harmonics for symmetric groups. Evidence suggests that these results fit into a larger framework in which Nakajima's quiver varieties play the role of the Hilbert scheme. In his continuing research, Haiman hopes to establish the validity of this larger framework and further develop the connections between combinatorics, Macdonald polynomials and Hilbert schemes. He also plans to work on unsolved problems connected with q-analogs of Littlewood-Richardson coefficients introduced by Shimozono and Weyman and the q-Schur functions of Lascoux, Lapointe and Morse. Finally, in collaboration with Grojnowski, he will return to a study he initiated some years ago of Hecke algebra characters and their connection with combinatorial properties of immanants. Combinatorics is the mathematical study of things that can be described, counted and manipulated in simple and explicit terms: things like trees, Young diagrams, or strings of symbols, that we can write on paper or encode in a computer. In my view, what makes combinatorics interesting is that deeply abstract mathematical concepts--such as geometric spaces in many dimensions, or algebras of symmetries--tend to have an underlying combinatorial structure. My current work connects the geometric properties of special spaces such as Hilbert schemes and quiver varieties with the combinatorial properties of Young diagrams and symmetric polynomials. By understanding the connection, we can deal with these abstract spaces on a concrete level and get important new insights about both the geometry of the spaces and the rules governing the combinatorics associated with them.