By Naihuan Jing

Algebraic combinatorics has developed into some of the most energetic components 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 in addition algebraic geometry, harmonic research and mathematical physics.

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

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2. Let y\ < ■ ■ ■ < yn-k and Zk > ■ ■ ■ > z\ be sequences of integers that are complementary in { 1 , . . , n } . Assume that k is even. Then in W we have k l([yi,y2,--- ,yn-k,Zk,zk-i,--- ,zi\) = ^T(n-ZJ). The barred permutations of this type form the poset, denoted W*, of the minimal length left coset representatives of Sn in W. Our terminology and all unexplained notation concerning partitions will follow [Ma]. We set p(k) := (k,k — l,... , 1), a "triangular partition" of length k. DIVIDED DIFFERENCES OF TYPE D 35 Given a strict partition a = (oti > • • • > ati > 0) C p(n — 1), we set a+ := (on + 1, a2 + 1, • • • , « ; + 1) if Z is even, and a+ := (ai + l , a 2 + 1,.

We need one more definition (cf. 3. 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.

Using the braid relations in W one easily shows that if ro £ R(w\), then, after breaking a ribbon in D, we get D' such that rr,/ g R(w\). In the case of the push down operation, it is clear that we get D' with ro — ru1 ■ Note that any configuration of boxes D C Da such that m e R(w\) can be o o obtained from D\ C Du by a sequence of operations of the above described two types. 9. ( "Maximal deformation" of D\ C D^) 42 HAIBAO DUAN AND PIOTR PRAGACZ • Pick the lowest ribbon. Push it down as many times as possible.