2017/07/11 by Dinakar Muthiah, Muthiah, Dinakar, Daniel L. Orr +1
Mathematics · #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Advanced Algebra and Geometry
paper · pdf · doi:10.48550/arxiv.1707.03366
We study walk algebras and Hecke algebras for Kac-Moody root systems. Each\nchoice of orientation for the set of real roots gives rise to a corresponding\n"oriented" basis for each of these algebras. We show that the notion of\ndistinguished subexpression naturally arises when studying the transition\nmatrix between oriented bases. We then relate these notions to the geometry of\nKac-Moody flag varieties and Bott-Samelson varieties. In particular, we show\nthat the number of points over a finite field in certain intersections of these\nvarieties is given by change of basis coefficients between oriented bases of\nthe Hecke algebra. Using these results we give streamlined derivations of\nDeodhar's formula for R-polynomials and point-counting formulas for\nspecializations of nonsymmetric Macdonald polynomials E_\λ(\q,t)\nat \q=0,\∞.\n