Algebraic combinatorics and quantum groups by Naihuan Jing

By Naihuan Jing

Algebraic combinatorics has developed into essentially the most lively parts of arithmetic over the past numerous many years. Its fresh advancements became extra interactive with not just its conventional box illustration conception but in addition algebraic geometry, harmonic research and mathematical physics.

This booklet offers articles from a few of the key participants within the quarter. It covers Hecke algebras, corridor algebras, the Macdonald polynomial and its deviations, and their relatives with different fields.

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Let us fix an extending vertex p of the graph Q. For every a = d i m F let us fix the Nakajima's character: 0(V) = —1, if dim V = k € I' =1- {p}, and 6(V) = £ < 6 / , d i m c ( ^ ) if dimV = p e I (cf. [Nak98]). (Semi)stable points in this section are considered with respect to this sta­ bility condition. e. the extending vertex is a sink and one can get to the extending vertex from any other point in the quiver going along the oriented edges. In the DE case there is only one such orientation (once the extending vertex is fixed).

Every D-box in the i-th column gives us s,. ~D-boxes give no contribution. Then, rr> is the word obtained by writing the resulting s^ 's from right to left. ) For example, for n = 9 and /J = (8, 7,4,2), 9 8 7 6 5 4 3 2 1 d^ = ds o SQ o dy o dg o 83 o d4 o ss o 8Q o S7 o ss o 02 o d^ o S4 o 9 5 OSQ O r& S7 o Sg = S6 ■ S5 ■ S7 ■ SS ■ S4 ■ S6 ■ S7 ■ Sg . o One can easily prove that for D = D^, we have ro € R{wn) - the set of reduced decompositions of w^. This is our distinguished reduced decomposition of wM.

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