You are here

Algebraic combinatorics and quantum groups by Naihuan Jing

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.

Show description

Read or Download Algebraic combinatorics and quantum groups PDF

Similar algebraic geometry books

Fukaya Categories and Picard-Lefschetz Theory

The imperative items within the ebook are Lagrangian submanifolds and their invariants, equivalent to Floer homology and its multiplicative buildings, which jointly represent the Fukaya classification. The appropriate points of pseudo-holomorphic curve concept are coated in a few aspect, and there's additionally a self-contained account of the mandatory homological algebra.

Additional info for Algebraic combinatorics and quantum groups

Example text

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.

Download PDF sample

Rated 4.00 of 5 – based on 22 votes