 By Naihuan Jing

Algebraic combinatorics has advanced into probably the most lively components of arithmetic over the last numerous many years. Its fresh advancements became extra interactive with not just its conventional box illustration concept but additionally algebraic geometry, harmonic research and mathematical physics.

This ebook provides articles from the various key individuals within the zone. It covers Hecke algebras, corridor algebras, the Macdonald polynomial and its deviations, and their family with different fields.

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Additional resources for Algebraic combinatorics and quantum groups

Example text

N, and if k denotes the largest number such that i\, < n, then n — k is even. Let us denote by fl[z'i,... ,ik] the Schubert variety in G determined (via restriction to G) by this Schubert variety in Gn(U), that is: n[h,... ,ifc] :={LeG:dim(LnVip) > p V p = 1 , . . ,fc}. t. the ordered basis {i>i,... ) One has dim Cl[ii,... , ik) = h H + ik + n(n - k) - n(n + l ) / 2 . The Schubert classes in H* (G) determined by these Schubert varieties are related in following way to the Schubert classes aa considered earlier in this section.

The Grassmannian of complex structures parametrizes orthogonal auto­ morphisms of the Euclidean space R 2 n whose square is the minus identity. Equivalently, it parametrizes minimal geodesies from the identity to the mi­ nus identity in the orthogonal group 5 0 ( 2 n , R ) [Mi]. This space is usually denoted by CSn. It played a significant role in several important achieve­ ments in topology: 31 32 HAIBAO DUAN AND PIOTR PRAGACZ • in the investigation of orthonormal vector fields on spheres by Hurewicz and Adams; • in the study of the existence of complex structures on even dimen­ sional spheres by Borel and Serre; • in the Bott's discovery of the eight-periodicity of homotopy groups of the stable real orthogonal groups.

T. dime L = 1, A(LR) = LR and A\LR = J0\LR}. t.