Download Algebraic combinatorics and quantum groups by Naihuan Jing PDF

By Naihuan Jing

Algebraic combinatorics has developed into the most energetic components of arithmetic over the past a number of a long time. 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 a number of the key individuals within the sector. It covers Hecke algebras, corridor algebras, the Macdonald polynomial and its deviations, and their relatives with different fields.

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Let now / i b e a strict partition with even length. (E) = 0 (we say: " D causes the vanishing"). In many computations in this section, we will apply compositions of the operators of boxes of £>p to the generating functions Ea. With the following example we illustrate how such operators act. 4. Let n = 9. We apply the operators of boxes from left to right to Ea, where a = (ag, as, a7, a$, a$, a4, a3, a 2 ,01), and obtain E^; "• " denotes a D-box and empty boxes are ~D-boxes. We give 2 examples of the action of the operators associated with a row in D^: 9 8 7 6 5 4 3 2 1 l*lxl*l*l*l*l*l*l a 1 ' = (-a&,a7,ae,as,a4,a3:a2,0,0), 9 8 7 6 5 4 3 2 1 lxl*l*l*l 1*1*1 1*1 a'= (as,a7,a6,0,a4,a3,0,ai,0).

N-l, di(Ea) = d-Ea4f ai=ai+1+d (d = - 2 , - 1 , 0 , 1 , 2 ) , where a! = ( a n , . . , 0 , 0 , . . , a,\) is the sequence a with a^+i, a* replaced by zeros. ,a1)In particular if A is a composition of some s- and d-operations, then for every a, A(E&) = (scalar) ■ E&', where a' is uniquely determined if this scalar is not zero. Proof. g. (c). We have, with a' = (0,0, o„_2, • • • , a\), On\&&) 0)(i (1 + + anxn) - (1 - an-ixn){l = - anxn-i) — Xn—1 ~r Xn • &&' (q n _! + a n )(a: ra -i +x r a ) — — Xn—\ -f- Xn • £>a!

4. A G R O U P - T H E O R E T I C APPROACH TO SCHUBERT CALCULUS FOR CSn We first introduce some notation. Recall that H = S0(2n,C) and B C H is a Borel subgroup of H. We denote by P - the maximal par­ abolic subgroup of H containing B and corresponding to the subset S of HAIBAO DUAN AND PIOTR PRAGACZ 46 simple roots minus the right end root, by F - an "isotropic" orthogonal flag manifold H/B, and by G - the orthogonal Grassmannian H/P. Moreover, the Schubert variety Xw, w e W, is defined as the closure of the Schubert cell B~wB/B in H/B (B~ is the opposite Borel subgroup to B).

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