The Coinvariant Algebra in Positive Characteristic

Work with Victor Reiner and Dennis Stanton, available as a preprint from http://www.math.umn.edu/∼webb

Reflection Groups

Let V be a vector space over k. A linear map g : V → V is a reflection if and only if it has finite order and fixes pointwise a hyperplane. A finite group G is a reflection group if and only if G is a group of linear automorphisms of a vector space and is generated by reflections.

In characteristic zero, real reflection groups are Coxeter groups. Over the complex numbers they are given by the list of Shephard and Todd.

In positive characteristic reflections need not be diagonalizable. Transvections such as

( \begin{matrix}1 & & 0 & ∗ \\ & ... & & ∗ \\ 0 & & 1 & ∗ \\ 0 & & & 1 \end{matrix} )

are reflections. Any group generated by transvections is a reflection group, for example SL(V) and U(V) = upper uni-triangular matrices. Also GL(V) is a reflection group since

( \begin{matrix}λ \\ & 1 \\ & & . & . & . \\ & & & & & 1 \end{matrix} )

is a reflection. The permutation module for Σ_{n} is exhibits Σ_{n} as a reflection group.

Invariants

Let S = S(V^{∗}) be the symmetric algebra on V^{∗}, so choosing a basis x_{1}, …, x_{n} for V^{∗} we identify S as the polynomial ring k[x_{1}, …, x_{n}]. G acts on S by linear substitutions with invariants S^{G}.

THEOREM.

(1) (Shephard-Todd, Chevalley) If char k = 0 then V is a reflection group if and only if S^{G} is a polynomial ring.

(2) (Serre) Over any field, if S^{G} is a polynomial ring then G is a reflection group.

The converse of (2) is false and there is a standard example with the Weyl group of type F_{4}, see the book of Neusel and Smith.

We will assume always that S^{G} is polynomial, so for example over 𝔽_{p^{n}},G could be GL(V), a parabolic subgroup of GL(V), U(V), Σ_{n} acting on the permutation module for {1, …, n}, etc.

The coinvariant algebra

This is S ⊗_{S^{G}} k. Here we regard S as an S^{G}-module. Consider the ideal I ◁ S^{G} consisting of invariants of positive degree, so S^{G}/I ≅ k. Now S ⊗_{S^{G}} k ≅ S/S · I where S · I is the ideal in S generated by elements of S^{G} of positive degree. The notation S_{G} = S⊗_{S^{G}} k is sometimes used. There is an action of G on S ⊗_{S^{G}} k.

Example: Σ_{3} permutes x, y, z with invariants generated as a polynomial ring by the elementary symmetric polynomials in degrees 1, 2 and 3, namely e_{1} = x + y + z, e_{2} = xy + xz + yz, e_{3} = xyz.

If W is a Weyl group associated to a complex reductive algebra group G with Borel subgroup B, then H^{∗}(G/B) ≅ S ⊗ S^{W} ℂ.

THEOREM (Chevalley (char 0), Mitchell (char p)). If S^{G} is polynomial then S⊗_{S^{G}} k has the same composition factors as kG.

The two modules are isomorphic if and only if |G| is invertible in k. If S^{G} is polynomial then S is free as a S^{G}-module (since S is Cohen-Macaulay, so is free over any polynomial subring of the same dimension). S = S^{G}a_{1} ⊕ ⋯ ⊕ S^{G}a_{r} if and only if the images of a_{1}, …, a_{r} are a basis for S/S · I = S ⊗_{S^{G}} k.

We have Hilb(S, t) = Hilb(S^{G}, t) · Hilb(S ⊗_{S^{G}} k, t) so that

Hilb(S ⊗_{S^{G}} k, t) = \frac{Hilb(S, t)}{Hilb(S^{G}, t)}

= \frac{1}{(1 − t)^{n}} /\frac{1}{(1 − t^{d_{1}}) ⋯ (1 − t^{d_{n}})}

= (1 + t + ⋯ + t^{d_{1}−1}) ⋯ (1 + t + ⋯ + t^{d_{n}−1}) and we see that dim S ⊗_{S^{G}} k = d_{1} ⋯ d_{n}. In the case of a Coxeter group this polynomial is also known to be the Poincaré polynomial of the group.

Example. Σ_{4} acts on {w, x, y, z}. Hilb(S ⊗_{S^{G}} k, t) = (1 + t)(1 + t + t^{2})(1 + t + t^{2} + t^{3}).

Regular elements

Let k̄ be the algebraic closure of k. A vector v ∈ \overline{V} = V ⊗_{k} k̄ is regular if and only if it is not in any reflecting hyperplane of any reflection in G, if and only if v is fixed by no reflection in G. An element c ∈ G is regular if and only if it has an eigenvector which is regular.

Examples: (Springer, Invent. Math 25 (1974)) With Σ_{n} permuting x_{1}, …, x_{n} in characteristic zero the regular elements are the n-cycles and the (n−1)-cycles. (An eigenvector of any shorter cycle is fixed by an involution on two remaining points. An eigenvector of a permutation is an eigenvector of one of its cycles.)

LEMMA (Reiner-Stanton-Webb). If S^{G} is polynomial, char k = p and c is regular, then c is p-regular (i.e. has order prime to p).

Application: the Σ_{4} example in characteristics 2 and 3.

PROPOSITION (Reiner-Stanton-Webb). In GL_{n}(𝔽_{q}) the regular elements are exactly the powers of Singer cycles.

We define a new action of a regular element c ∈ G on \overline{S} := k̄ ⊗_{k} S. Let cv = ζv where ζ ∈ k̄^{×} and v is regular. The order of c is divisible by the order of ζ (and in fact these orders are equal when S^{G} is polynomial). Let c act on \overline{S}_{n} as multiplication by ζ^{n}. Thus c acts on \overline{S} ⊗\overline{S}^{G} k̄. The action commutes with that of G, and these become k̄[G × ⟨c⟩]- modules. We regard k̄G also as a k̄[G × ⟨c⟩]-module, where G acts by left multiplication and c acts by right multiplication (by c^{−1}).

v ∈ \overline{V} = V ⊗_{k} k̄ is regular if and only if it is not in any reflecting hyperplane of any reflection in G, if and only if v is fixed by no reflection in G. An element c ∈ G is regular if and only if it has an eigenvector which is regular.

Examples: (Springer, Invent. Math 25 (1974)) With Σ_{n} permuting x_{1}, …, x_{n} in characteristic zero the regular elements are the n-cycles and the (n−1)-cycles. (An eigenvector of any shorter cycle is fixed by an involution on two remaining points. An eigenvector of a permutation is an eigenvector of one of its cycles.)

LEMMA (Reiner-Stanton-Webb). If S^{G} is polynomial, char k = p and c is regular, then c is p-regular (i.e. has order prime to p).

Application: the Σ_{4} example in characteristics 2 and 3.

PROPOSITION (Reiner-Stanton-Webb). In GL_{n}(𝔽_{q}) the regular elements are exactly the powers of Singer cycles.

We define a new action of a regular element c ∈ G on \overline{S} := k̄ ⊗_{k} S. Let cv = ζv where ζ ∈ k̄^{×} and v is regular. The order of c is divisible by the order of ζ (and in fact these orders are equal when S^{G} is polynomial). Let c act on \overline{S}_{n} as multiplication by ζ^{n}. Thus c acts on \overline{S} ⊗\overline{S}^{G} k̄. The action commutes with that of G, and these become k̄[G × ⟨c⟩]- modules. We regard k̄G also as a k̄[G × ⟨c⟩]-module, where G acts by left multiplication and c acts by right multiplication (by c^{−1}).

THEOREM (Springer (char 0), Reiner-Stanton-Webb (char p)). As k̄[G × ⟨c⟩]- modules, \overline{S} ⊗_{S^{G}} k̄ and k̄G have the same composition factors.

Proof. This is a Brauer character calculation.

Application: k̄C is semisimple and k̄C = k̄Ce_{0} ⊕ ⋯ ⊕ k̄Ce_{d−1} where e_{i}^{2} = e_{i} and for any right k̄C-module M, Me_{i} is the ζ^{i} eigenspace of c. Thus k̄G = k̄Ge_{0} ⊕ ⋯ ⊕ k̄Ge_{d−1} is a direct sum of projective k̄G-modules. Since c acts freely on G, these summands all have the same dimension. On \overline{S} ⊗\overline{S}^{G} k̄, the ζ^{i}-eigenspace is the sum of the homogeneous terms in degrees r ≡ i (mod d). This sum has the same k̄G-module composition factors as k̄Ge_{i}.

THEOREM. ⨁_{r≡i (mod d)}(\overline{S} ⊗\overline{S}^{G} k̄)_{r} has the composition factors of a projective k̄Gmodule.

Question: Does there exist a filtration of k̄G by k̄[G × C]-modules so that the factors, taken in ascending order, are isomorphic as k̄[G × C]-modules to the homogeneous terms of \overline{S} ⊗\overline{S}^{G} k̄ taken in ascending order?