Strong Lefschetz Elements of the Coinvariant Rings of Finite Coxeter Groups

Abstract

For the coinvariant rings of finite Coxeter groups of types other than H 4 , we show that a homogeneous element of degree one is a strong Lefschetz element if and only if it is not fixed by any reflections. We also give the necessary and sufficient condition for strong Lefschetz elements in the invariant subrings of the coinvariant rings of Weyl groups.

Lefschetz elements of Artinian Gorenstein algebras and Hessians of homogeneous polynomials

“…In fact, the coinvariant algebra of the Weyl group is isomorphic to the cohomology ring of the corresponding flag variety. In [11] and [12], it has been shown that the coinvariant algebra of any finite Coxter group has the Lefschetz property and that the set of the Lefschetz elements is the complement of the union of the reflection hyperplanes except for type H 4 case. The determination of the set of the Lefschetz elements is still open for H 4 because of the computational complexity.…”

The strong Lefschetz property for coinvariant rings of finite reflection groups

“…It should be added that the hard Lefschetz theorem is highly non-trivial: Hodge proved it over C using his theory of harmonic integrals (now called Hodge theory) and later Deligne proved it in characteristic p using a version of the Riemann hypothesis for varieties over finite fields; see [12] for more details. For noncrystallographic W, Theorem 1 has been verified by direct computation in types I 2 (m) and H 3 by Maeno, Numata and Wachi [11] and in type H 4 by Numata and Wachi [13]. Here also it should be added that the computations for type H in [11] and [13] are very large and are carried out using the computer algebra package Macaulay2.…”

“…For noncrystallographic W, Theorem 1 has been verified by direct computation in types I 2 (m) and H 3 by Maeno, Numata and Wachi [11] and in type H 4 by Numata and Wachi [13]. Here also it should be added that the computations for type H in [11] and [13] are very large and are carried out using the computer algebra package Macaulay2.…”

The Lefschetz Elements of Coinvariant Algebras of Binary Polyhedral Groups

“…In general the coinvariant algebras of finite groups are not always isomorphic to the cohomology rings of any manifolds. Indeed the coinvariant algebras of non-crystallographic finite Coxeter groups are not isomorphic to the cohomology rings of any varieties, but have the Lefschetz properties ( [6], [9], [10], [13]). See [2], [3], [7], [13], [14], for the study of Lefschetz properties for wider classes of graded Artinian algebras.…”

“…x, y, x 2 , xy, y 2 , x 3 , x 2 y, xy 2 , y 3 , x4 , x 3 y, x 2 y 2 , xy3 , y 4 , x5 , x 4 y, x 3 y 2 , x 2 y 3 , xy4 , y 5 , x6 , x 5 y, x 4 y 2 , x 3 y 3 , x 2 y 4 , xy5 , y 6 , x7 , x 6 y, x 5 y 2 , x 4 y 3 , x 3 y 4 , x 2 y 5 , xy6 , y 7 , x 8 , x 7 y, x 6 y 2 , x 5 y 3 , x 4 y 4 , x 3 y 5 , x 2 y 6 , xy7 , y 8 , x9 , x 8 y, x 7 y 2 , x 6 y 3 , x 5 y 4 , x 4 y 5 , x 3 y 6 , x 2 y 7 , xy8 , y 9 , x 10 , x 9 y, x 8 y 2 , x 7 y 3 , x 6 y 4 , x 5 y 5 , x 4 y 6 , x 3 y 7 , x 2 y 8 , xy9 , y 10 , x11 , x 10 y, x 9 y 2 , x 8 y 3 , x 7 y 4 , x 6 y 5 , x 5 y 6 , x 4 y 7 , x 3 y 8 , x 2 y 9 , xy 10 , y11 , x12 , x 11 y, x 10 y 2 , x 9 y 3 , x 8 y 4 , x 7 y 5 , x 5 y 7 , x 4 y 8 , x 3 y 9 , x 2 y 10 , xy 11 , y 12 , x13 , x 12 y, x 11 y 2 , x 10 y 3 , x 9 y 4 , x 8 y 5 , x 5 y 8 , x 4 y 9 , x 3 y 10 , x 2 y 11 , xy12 , y 13 , x14 , x 13 y, x 12 y 2 , x 11 y 3 , x 10 y 4 , x 9 y 5 , x 5 y 9 , x 4 y 10 , x 3 y 11 , x 2 y 12 , xy13 , y14 , x 15 , x 14 y, x 13 y 2 , x 12 y 3 , x 11 y 4 , x 10 y 5 , x 5 y 10 , x 4 y 11 , x 3 y 12 , x 2 y 13 , xy 14 , y 15 , x 16 , x 15 y, x 14 y 2 , x 13 y 3 , x 12 y 4 , x 11 y 5 , x 5 y 11 , x 4 y 12 , x 3 y 13 , x 2 y 14 , xy 15 , y 16 , x 17 , x 16 y, x 15 y 2 , x 14 y 3 , x 13 y 4 , x 12 y 5 , x 5 y 12 , x 4 y 13 , x 3 y 14 , x 2 y 15 , xy 16 , y 17 , x 18 , x 17 y, x 16 y 2 , x 15 y 3 , x 14 y 4 , x 13 y 5 , x 5 y 13 , x 4 y 14 , x 3 y 15 , x 2 y 16 , xy 17 , y 18 , x 19 , x 18 y, x 17 y 2 , x 16 y 3 , x 15 y 4 , x 14 y 5 , x 5 y 14 , x 4 y 15 , x 3 y 16 , x 2 y 17 , xy 18 , y 19 , x 19 y, x 18 y 2 , x 17 y 3 , x 16 y 4 , x 15 y 5 , x 5 y 15 , x 4 y 16 , x 3 y 17 , x 2 y 18 , xy 19 , y 20 , x 20 y, x 19 y 2 , x 18 y 3 , x 17 y 4 , x 16 y 5 , x 5 y 16 , x 4 y 17 , x 3 y 18 , x 2 y 19 , xy 20 , x 20 y 2 , x 19 y 3 , x 18 y 4 , x 17 y 5 , x 5 y 17 , x 4 y 18 , x 3 y 19 , x 2 y 20 , xy 21 , x 21 y 2 , x 20 y 3 , x 19 y 4 , x 18 y 5 , x 5 y 18 , x 4 y 19 , x 3 y 20 , x 2 y 21 , x 21 y 3 , x 20 y 4 , x 19 y 5 , x 5 y 19 , x 4 y 20 , x 3 y 21 , x 2 y 22 , x 22 y 3 , x 21 y 4 , x 20 y 5 , x 5 y 20 , x 4 y 21 , x 3 y 22 , x 22 y 4 , x 21 y 5 , x 5 y 21 , x 4 y 22 , x 3 y 23 , x 23 y 4 , x 22 y 5 , x 5 y 22 , x 4 y 23 , x 23 y 5 , x 5 y 23 , x 4 y 24 , x 24 y 5 , x 5 y 24 , x 5 y 25 (8.0) The element l = ax +by ∈ S 1 (V ) G is the strong Lefschetz element if and only if 21 i=1g i = 0, where g i is a polynomial in a, b given by g 1 (a, b) = a , g 2 (a, b) = b , g 3 (a, b) = a 2 + ab − b 2 , g 4 (a, b) = a 2 + b 2 , g 5 (a, b) = a 4 − 3a 3 b + 4a 2 b 2 − 2ab 3 + b 4 , g 6 (a, b) = a 4 + 2a 3 b + 4a 2 b 2 + 3ab 3 + b 4 , g 7 (a, b) = a 4 − 3a 3 b − a 2 b 2 + 3ab 3 + b 4 , g ...…”