2 research outputs found
A Class of Kazhdan-Lusztig R-Polynomials and q-Fibonacci Numbers
Let denote the symmetric group on . For two
permutations such that in the Bruhat order, let
and denote the Kazhdan-Lusztig -polynomial and
-polynomial, respectively. Let , and let be a
permutation such that . We obtain a formula for the
-polynomials in terms of the -Fibonacci numbers
depending on a parameter determined by the reduced expression of . When
is the identity , this reduces to a formula obtained by Pagliacci.
In another direction, we obtain a formula for the -polynomial
, where . In a more general context, we conjecture that for any two permutations
such that , the -polynomial
can be expressed as a product of -Fibonacci numbers
multiplied by a power of .Comment: 11 page
Diagrammatics for Kazhdan-Lusztig R-polynomials
Let be an arbitrary Coxeter system. We introduce a family of
polynomials, , indexed by pairs
formed by an element and a (non-necessarily
reduced) word in the alphabet . The polynomial
is obtained by considering a certain
subset of Libedinsky's light leaves associated to the pair .
Given a reduced expression of an element ; we show that
coincides with the Kazhdan-- Lusztig
-polynomial . Using the diagrammatic approach,
we obtain some closed formulas for - polynomials.Comment: 20 pages, best viewed in colo