### Abstract

Let rho : G hooked right arrow GL(n,F) be a representation of a finite group G over a finite field F and f(1),..., f(n) epsilon F[V](G) such that the ring of invariants is a polynomial algebra F[f(1),. . . ,f(n)]. It is known that in this case the algebra of coinvariants F[V](G) is a Poincare duality algebra, and if, moreover, the order of G is invertible in F, that a fundamental class is represented by the Jacobian determinant det[partial derivativef(i)/partial derivativez(j)], and is therefore a det(-1)-relative invariant. In this note we deduce what happens in the modular case. As a bonus we obtain a new criterion for an unstable algebra over the Steenrod algebra to be a ring of invariants.

Original language | English |
---|---|

Pages (from-to) | 597-611 |

Number of pages | 15 |

Journal | Proceedings of the Edinburgh Mathematical Society |

Volume | 44 |

Publication status | Published - 2001 |

### Keywords

- rings of coinvariants
- reflection groups
- Steenrod operations
- Jacobian determinants
- Poincare duality algebras
- relative invariants
- RINGS
- ALGEBRAS
- THEOREM

### Cite this

**Invariants and coinvariants of finite pseudoreflection groups, Jacobian determinants and Steenrod operations.** / Smith, L .

Research output: Contribution to journal › Article

*Proceedings of the Edinburgh Mathematical Society*, vol. 44, pp. 597-611.

}

TY - JOUR

T1 - Invariants and coinvariants of finite pseudoreflection groups, Jacobian determinants and Steenrod operations

AU - Smith, L

PY - 2001

Y1 - 2001

N2 - Let rho : G hooked right arrow GL(n,F) be a representation of a finite group G over a finite field F and f(1),..., f(n) epsilon F[V](G) such that the ring of invariants is a polynomial algebra F[f(1),. . . ,f(n)]. It is known that in this case the algebra of coinvariants F[V](G) is a Poincare duality algebra, and if, moreover, the order of G is invertible in F, that a fundamental class is represented by the Jacobian determinant det[partial derivativef(i)/partial derivativez(j)], and is therefore a det(-1)-relative invariant. In this note we deduce what happens in the modular case. As a bonus we obtain a new criterion for an unstable algebra over the Steenrod algebra to be a ring of invariants.

AB - Let rho : G hooked right arrow GL(n,F) be a representation of a finite group G over a finite field F and f(1),..., f(n) epsilon F[V](G) such that the ring of invariants is a polynomial algebra F[f(1),. . . ,f(n)]. It is known that in this case the algebra of coinvariants F[V](G) is a Poincare duality algebra, and if, moreover, the order of G is invertible in F, that a fundamental class is represented by the Jacobian determinant det[partial derivativef(i)/partial derivativez(j)], and is therefore a det(-1)-relative invariant. In this note we deduce what happens in the modular case. As a bonus we obtain a new criterion for an unstable algebra over the Steenrod algebra to be a ring of invariants.

KW - rings of coinvariants

KW - reflection groups

KW - Steenrod operations

KW - Jacobian determinants

KW - Poincare duality algebras

KW - relative invariants

KW - RINGS

KW - ALGEBRAS

KW - THEOREM

M3 - Article

VL - 44

SP - 597

EP - 611

JO - Proceedings of the Edinburgh Mathematical Society

JF - Proceedings of the Edinburgh Mathematical Society

SN - 0013-0915

ER -