𝔖 Bobbio Scriptorium
✦   LIBER   ✦

On the Ring of Simultaneous Invariants for the Gleason–MacWilliams Group

✍ Scribed by Eiichi Bannai; Etsuko Bannai; Michio Ozeki; Yasuo Teranishi

Publisher
Elsevier Science
Year
1999
Tongue
English
Weight
128 KB
Volume
20
Category
Article
ISSN
0195-6698

No coin nor oath required. For personal study only.

✦ Synopsis

We construct a canonical generating set for the polynomial invariants of the simultaneous diagonal action (of arbitrary number of l factors) of the two-dimensional finite unitary reflection group G of order 192, which is called the group No. 9 in the list of Shephard and Todd, and is also called the Gleason-MacWilliams group. We find this canonical set in the vector space (⊗ l i=1 V ) G , where V denotes the (dual of the) two-dimensional vector space on which the group G acts, by applying the techniques of Weyl (i.e., the polarization process of invariant theory) to the invariants C[x, y] G 0 of the two-dimensional group G 0 of order 48 which is the intersection of G and S L(2, C). It is shown that each element in this canonical set corresponds to an irreducible representation which appears in the decomposition of the action of the symmetric group S l . That is, by letting the symmetric group S l acts on each element of the canonical generating set, we get an irreducible subspace on which the symmetric group S l acts irreducibly, and all these irreducible subspaces give the decomposition of the whole space (⊗ l i=1 V ) G . This also makes it possible to find the generating set of the simultaneous diagonal action (of arbitrary l factors) of the group G. This canonical generating set is different from the homogeneous system of parameters of the simultaneous diagonal action of the group G. We can construct Jacobi forms (in the sense of Eichler and Zagier) in various ways from the invariants of the simultaneous diagonal action of the group G, and our canonical generating set is very fit and convenient for the purpose of the construction of Jacobi forms.

📜 SIMILAR VOLUMES

Rings of Invariants of Certainp-Groups o
Rings of Invariants of Certainp-Groups over the FieldFp
✍ H.E.A. Campbell; I.P. Hughes 📂 Article 📅 1999 🏛 Elsevier Science 🌐 English ⚖ 135 KB

We use Nakajima's J. Algebra 85 1983 , 253᎐286 characterization of p-groups with polynomial rings of invariants over the field F to prove that maximal proper p subgroups of such groups have rings of invariants that are hypersurfaces. We give an explicit construction for the ring of invariants of suc

Subgroup Theorems for the Baer-Invariant
Subgroup Theorems for the Baer-Invariant of Groups
✍ M.R.R. Moghaddam; B. Mashayekhy; S. Kayvanfar 📂 Article 📅 1998 🏛 Elsevier Science 🌐 English ⚖ 193 KB

that if G is a finite group with a subgroup H of finite index n, then the nth power Ž . n Ž . of the Schur multiplier of G, M G , is isomorphic to a subgroup of M H . In this paper we prove a similar result for the centre by centre by w variety of groups, where w is any outer commutator word. Then u

Algebra Structure on the Hochschild Coho
Algebra Structure on the Hochschild Cohomology of the Ring of Invariants of a Weyl Algebra under a Finite Group
✍ Mariano Suarez Alvarez 📂 Article 📅 2002 🏛 Elsevier Science 🌐 English ⚖ 139 KB

group of linear automorphisms of A . In this paper, we compute the multiplicative n ⅷ Ž G . structure on the Hochschild cohomology HH A of the algebra of invariants of n ⅷ Ž G . G. We prove that, as a graded algebra, HH A is isomorphic to the graded n algebra associated to the center of the group al

On the Cohen–Macaulay Property of Modula
On the Cohen–Macaulay Property of Modular Invariant Rings
✍ Gregor Kemper 📂 Article 📅 1999 🏛 Elsevier Science 🌐 English ⚖ 203 KB

If V is a faithful module for a finite group G over a field of characteristic p, then the ring of invariants need not be Cohen᎐Macaulay if p divides the order of G. In this article the cohomology of G is used to study the question of Cohen᎐Macaulayness of the invariant ring. One of the results is a

Generating a Noetherian Normalization of
Generating a Noetherian Normalization of the Invariant Ring of a Finite Group
✍ Wolfram Decker; Agnes Eileen Heydtmann; Frank-Olaf Schreyer 📂 Article 📅 1998 🏛 Elsevier Science 🌐 English ⚖ 320 KB

We improve Kemper's algorithm for the computation of a noetherian normalization of the invariant ring of a finite group. The new algorithm, which still works over arbitrary coefficient fields, no longer relies on algorithms for primary decomposition.

The Hit Problem for the Modular Invarian
The Hit Problem for the Modular Invariants of Linear Groups
✍ Nguyen H.V Hu·ng; Tran Ngoc Nam 📂 Article 📅 2001 🏛 Elsevier Science 🌐 English ⚖ 176 KB

Let the mod 2 Steenrod algebra, , and the general linear group, GL k = GL k 2 , act on P k = 2 x 1 x k with deg x i = 1 in the usual manner. We prove that, for a family of some rather small subgroups G of GL k , every element of positive degree in the invariant algebra P G k is hit by in P k . In ot