By Naihuan Jing

Algebraic combinatorics has developed into essentially the most energetic components of arithmetic over the last numerous a long time. Its contemporary advancements became extra interactive with not just its conventional box illustration concept but in addition algebraic geometry, harmonic research and mathematical physics.

This publication offers articles from the various key individuals within the quarter. It covers Hecke algebras, corridor algebras, the Macdonald polynomial and its deviations, and their kinfolk with different fields.

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**Sample text**

11. 267] and in more detail [HH]) is the proof of the following equality: for a strict partition 7 = (71 > • • • > 7/ > 0),

Since the functions E^,Ek(m) are constant on GV-orbits, we can con sider them as functions on Ay/Gv- The equalities in the following conjec ture are understood in this sense. Conjecture. 1) Up to a sign, £ £ | A J , = Ea, dimV = a £ R+. 2) Up to a sign, 2%(l)|A(, = Ek{\), dim V = S. 3) Let d i m F = m5, m > 1. We conjecture that if x',x" £ Assd are two elements in the same S-equivalence class and Ek(m)(x') ^ 0, E*k{m){x") ^ 0, then E*k(m)(x') = E*k(m){x"). »d as a function on Assd/S-equivalence by setting for an S-equivalence class X: fr*{ \(v\ _ J Ek(m)(x)> 10, */ there exists x £ X with Ek(m){x) otherwise.

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. For a strict partition a = (ai,... , afc) C p(n — 1), one has aa = f2[n — ot\,...