Comparing Codimension and Absolute Length in Complex Reflection Groups

Abstract

Reflection length and codimension of fixed point spaces induce partial orders on a complex reflection group. Motivated by connections to the algebraic structure of cohomology governing deformations of skew group algebras, we show that Coxeter groups and the infinite family G(m, 1, n) are the only irreducible complex reflection groups for which reflection length and codimension coincide. We then discuss implications for the degrees of generators of Hochschild cohomology. Along the way, we describe the codimension atoms for the infinite family G(m, p, n), give algorithms using character theory, and determine two-variable Poincaré polynomials recording reflection length and codimension.

Key Words:

• Codimension
• Hochschild cohomology
• Partial orders
• Reflection groups
• Reflection length

2010 Mathematics Subject Classification:

• 05E15
• 20F55
• 16E40
• 06A06

ACKNOWLEDGMENTS

The author thanks her Ph.D. advisor Anne Shepler for suggesting this project and for helpful discussions. The author also thanks Cathy Kriloff for suggestions on an early draft.

Notes

Brady and Watt [5] prove ≤_⊥ is a partial order. Their proof is also valid when codimension is replaced by any function μ: G → [0, ∞) satisfying μ(a) = 0 iff a = 1 (positive definite) and μ(ab) ≤ μ(a) + μ(b) for all a, b in G (subadditive).

Note that the factor dv _g ∧ dv _h in Equation (7.4) may not a priori be an element of . To interpret the equation correctly, we must apply to the wedge product dv _g ∧ dv _h the projection induced by the orthogonal projection V* → (V ^gh )*. After the last iteration of the cup product formula, we also apply the projections S(V) → S(V)/I((V ^g )^⊥) ≅ S(V ^g ) to the polynomial parts to obtain a representative in HH^•(S(V), S(V)#G.

Communicated by E. Kirkman.