Algebraic combinatorics and quantum groups by Naihuan Jing

By Naihuan Jing

Algebraic combinatorics has advanced into the most energetic parts of arithmetic over the past a number of a long time. Its contemporary advancements became extra interactive with not just its conventional box illustration thought but additionally algebraic geometry, harmonic research and mathematical physics.

This booklet offers articles from many of the key individuals within the sector. It covers Hecke algebras, corridor algebras, the Macdonald polynomial and its deviations, and their kin with different fields.

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Example text

These are the so called P-functions of [PR1], which are modeled on Schur P-functions. In [LP], Lascoux and the second author worked out a connection of orthogonal divided differences to P-functions using vertex operators. This has led to orthogonal Schubert polynomials that are useful in various cohomological computations (cf. a recent work of Kresch and Tamvakis [KT,T2], and Buch [BKT]). After presenting the Schubert varieties in a group-theoretic way, we also describe them via Schubert-type conditions relative to some flag of linear subspaces, and finally we study them in terms of complex structures.

Xn), i = 1 , . . ,n. 1. (a) We have Si(Ea) = Ea>, where 1 _ («n. • • • ,aj + 2,ai,

More open questions. 5) with respect to semicanonical basis make sense for any affine Lie algebra of type ADE. It would be very interesting to develop a purely combinatorial approach to these questions in the affine case as well. REFERENCES [CB01] [DFR00] [FM00] [FK81] [FMV01] [Gab72] [HM98] [KVOO] [KS97] [KL88] [King94] [Kron89] [Lus90] [Lus91] [Lus92] W. Crawley-Boevey, Geometry of the moment map for representations of quiv­ ers, Compositio Math. 126 (2001), no. 3, 257-293. M. Douglas, B. Fiol, and C.

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