Star reducible Coxeter groups

Details Star reducible Coxeter groups Star reducible Coxeter groups

2005-09-15

arxiv.org/abs/math/0509363v2

Mathematics

Quantum Algebra

Approximately 41 pages, AMSTeX, 4 figures. Revised in light of referee comments. To appear in the Glasgow Mathematical Journal

Scientific paper

We define ``star reducible'' Coxeter groups to be those Coxeter groups for which every fully commutative element (in the sense of Stembridge) is equivalent to a product of commuting generators by a sequence of length-decreasing star operations (in the sense of Lusztig). We show that the Kazhdan--Lusztig bases of these groups have a nice projection property to the Temperley--Lieb type quotient, and furthermore that the images of the basis elements $C'_w$ (for fully commutative $w$) in the quotient have structure constants in ${\Bbb Z}^{\geq 0}[v, v^{-1}]$. We also classify the star reducible Coxeter groups and show that they form nine infinite families (types $A_n$, $B_n$, $D_n$, $E_n$, $F_n$, $H_n$, affine $A_{n-1}$ for $n$ odd, affine $C_{n-1}$ for $n$ even, and the case where the Coxeter graph is complete), with two exceptional cases (of ranks 6 and 7). This paper is the sequel to math.QA/0509362.

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