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Walk algebras, distinguished subexpressions, and point counting in Kac-Moody flag varieties
We study walk algebras and Hecke algebras for Kac-Moody root systems. Each
choice of orientation for the set of real roots gives rise to a corresponding
"oriented" basis for each of these algebras. We show that the notion of
distinguished subexpression naturally arises when studying the transition
matrix between oriented bases. We then relate these notions to the geometry of
Kac-Moody flag varieties and Bott-Samelson varieties. In particular, we show
that the number of points over a finite field in certain intersections of these
varieties is given by change of basis coefficients between oriented bases of
the Hecke algebra. Using these results we give streamlined derivations of
Deodhar's formula for -polynomials and point-counting formulas for
specializations of nonsymmetric Macdonald polynomials
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