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We introduce and study equivariant Hilbert series of ideals in polynomial rings in countably many variables that are invariant under a suitable action of a symmetric group or the monoid $Inc(\mathbb{N})$ of strictly increasing functions. Our first main result states that these series are rational functions in two variables. A key is to introduce also suitable submonoids of $Inc(\mathbb{N})$ and to compare invariant filtrations induced by their actions. Extending a result by Hillar and Sullivant, we show that any ideal that is invariant under these submonoids admits a Gr\"obner basis consisting of finitely many orbits. As our second main result we prove that the Krull dimension and multiplicity of ideals in an invariant filtration grow eventually linearly and exponentially, respectively, and we determine the terms that dominate this growth., Comment: 30 pages; the title has been modified

The gametization process reduces the study of non-commutative and non-associative algebras satisfying non-homogeneous polynomial identities with variables in X = { x 1 , … , x n } to algebras verifying simpler identities. However after a gametization, certain identities remain invariant and other identities, said universal invariant, are invariant for every gametization. Now in the case n = 1 , for all algebras satisfying a universal invariant polynomial identity studied until now, we know that the existence of an idempotent is not certain. Using an action of the gametization operators group on the non-commutative and non-associative algebra of polynomials K 〈 X 〉 , we give all identities which are invariant and universal invariant by gametization.

Let R be a prime ring, C its extended centroid and RF (resp. Q) its left (resp. symmetric) Martindale quotient ring. Let δ be a σ-derivation of R, where σ is an automorphism of R. We show the equivalence of K-polynomials (resp. K-identities) of δ and cv-polynomials (resp. semi-invariant polynomials) in the Ore extension Q[X;σ,δ]. We prove the existence of K-polynomials of δ in certain rather general family of maps. As applications, the following are proved among other things: Consider the expression ϕ(x):=∑i=0naiδi(x), where ai∈RF and an≠0.(1)If dimCϕ(R)C

Cataloged from PDF version of article. We consider a finite dimensional modular representation V of a cyclic group of prime order p. We show that two points in V that are in different orbits can be separated by a homogeneous invariant polynomial that has degree one or p and that involves variables from at most two summands in the dual representation. Simultaneously, we describe an explicit construction for a separating set consisting of polynomials with these properties. (C) 2009 Elsevier Inc. All rights reserved.

Let G be a finite group, k a perfect field, and V a finite-dimensional kG-module. We let G act on the power series k 〚 V 〛 by linear substitutions and address the question of when the invariant power series k 〚 V 〛 G form a unique factorization domain. We prove that for a permutation module for a p-group in characteristic p, the answer is always positive. On the other hand, if G is a cyclic group of order p, k has characteristic p, and V is an indecomposable kG-module of dimension r with 1 ⩽ r ⩽ p , we show that the invariant power series form a unique factorization domain if and only if r is equal to 1, 2, p − 1 or p. This contradicts a conjecture of Peskin.

Let the finite group G act linearly on the n-dimensional vector space V over the field k of characteristic p⩾0. Suppose H⊲G is a normal subgroup of index ℓ, a prime number. In this paper we shall study the relationship between the two invariant rings k[V]H and k[V]G. As a corollary of our main result we get that if k[V]G is a polynomial algebra and k[V]H is factorial then k[V]H is a graded hypersurface algebra.

For L a finite non-modular group whose invariants form a polynomial ring and H a subgroup of L containing the derived group of L, Nakajima found necessary and sufficient conditions on H for its invariant ring S H to be a hypersurface. In a crucial step of his proof he showed that if S H is a hypersurface, then between H and L there is a group G with polynomial invariant ring such that S H = S G [ b ] . For G a finite modular p-group over F p with polynomial invariant ring and H a subgroup of G containing the derived group of G, we find necessary and sufficient conditions on H to ensure that S H = S G [ b ] .

Let R be a prime ring, C its extended centroid and R F (resp. Q) its left (resp. symmetric) Martindale quotient ring. Let δ be a σ-derivation of R, where σ is an automorphism of R. We show the equivalence of K-polynomials (resp. K-identities) of δ and cv-polynomials (resp. semi-invariant polynomials) in the Ore extension Q [ X ; σ , δ ] . We prove the existence of K-polynomials of δ in certain rather general family of maps. As applications, the following are proved among other things: Consider the expression ϕ ( x ) : = ∑ i = 0 n a i δ i ( x ) , where a i ∈ R F and a n ≠ 0 . (1) If dim C ϕ ( R ) C ∞ then either R is a GPI-ring or ϕ ( x ) = 0 for all x ∈ R F . (2) If ϕ ( R ) ⊆ C then either R is commutative or ϕ ( x ) = 0 for all x ∈ R F .

Based on the first fundamental theorem of classical invariant theory we present a reduction technique for computing relative invariants for quivers with relations. This is applied to the invariant theory of canonical algebras and yields an explicit construction of the moduli spaces (together with the quotient morphisms from the corresponding representation spaces) for families of modules with a fixed dimension vector belonging to the central sincere separating subcategory. By means of a tilting process we extend these results to the invariant theory of concealed-canonical algebras, thus covering the cases of tame hereditary, tame concealed, and tubular algebras, respectively. Our approach yields, in particular, a uniform treatment to an essential part of the invariant theory of extended Dynkin quivers, a topic popular over the years, but stretches far beyond since also concealed-canonical algebras of tubular or wild representation type are covered.

In this paper we consider certain doubly transitive permutation groups which have monomial representations which are reducible. These groups have been studied by Hale and Shult 14) and by Taylor [ I7 1. The stabilizer H of a vertex has a subgroup L of index two for which elements of L are not conjugate in G to elements of H-L. The monomial representations have two invariant subspaces which can be reduced modulo any prime to form codes over finite fields. The invariant subspaces form global codes. This approach is motivated by the work of Ward 1 181 who considered the groups RSLf2, q) and Sp(2n, 4). The codes for PSL(2, q) give the classical extended quadratic residue codes. In Section 2, we construct the monomial representations and determine the two invariant subspaces. Taylor constructed 2-graphs from the centralizer algebra of these representations. The invariant subspaces are spanned by vectors whose entries are parameters in these Z&graphs. We illustrate the construction in the remaining sections with the doubiv transitive permutation representation on 176 vertices of the IIigman-Sims group. This permutation representation was first discovered by Graham Higman. The stabilizer of a vertex is a group H isomorphic to PTU(3, 5). It

In an earlier paper 121 the author discussed the relationship between invariants of the group G = Z/pZ in characteristic p and the classical covariants of SL(2. k) in characteristic zero. This is more closely studied here in the corresponding representation rings. In characteristic zero the only indecomposable (1 irreducible) SL(2, k)-modules are the R,: sI where R, is the set of homogeneous forms of degree n of the polynomial ring R = k/.x, y]. In characteristic p there are p indecomposable G-modules I’,,..., VP. The corresponding representation rings are

Let the mod 2 Steenrod algebra, A , and the general linear group, GLk := GL(k, F 2), act on Pk := F 2[x1, …, xk] with deg(xi) = 1 in the usual manner. We prove that, for a family of some rather small subgroups G of GLk, every element of positive degree in the invariant algebra PkG is hit by A in Pk. In other words, (PkG)+ ⊂ A + · Pk, where (PkG)+ and A + denote respectively the submodules of PkG and A consisting of all elements of positive degree. This family contains most of the parabolic subgroups of GLk. It should be noted that the smaller the group G is, the harder the problem turns out to be. Remarkably, when G is the smallest group of the family, the invariant algebra PkG is a polynomial algebra in k variables, whose degrees are ≤ 8 and fixed while k increases. It has been shown by Hu·ng [Trans. Amer. Math. Soc.349 (1997), 3893–3910] that, for G = GLk, the inclusion (PkGLk)+ ⊂ A + · Pk is equivalent to a weak algebraic version of the long-standing conjecture stating that the only spherical classes in Q0S0 are the elements of Hopf invariant 1 and those of Kervaire invariant 1.

6 discovered a remarkable checkerboard state model based on the Higman–Sims graph that yields a value of the Kauffman polynomial, which is a quantum invariant of links. We present a simple argument that the state model has the desired properties using the combinatorialB2spider 10 .

Hochschild and Mostow [3] have proved the finite generation of the algebra of invariants for the regular action of the unipotent radical of a parabolic subgroup on the polynomial functions on a complex semisimple algebraic group G. Pommerening [S] used combinatorial techniques to show that if G = SL, then we may remove any set of simple roots from the unipotent radical, and the algebra of invariants will still be finitely generated. In this paper we partially generalize Pommerening’s theorem to arbitrary semisimple groups G. In the proof we classify in terms of Dynkin diagrams, the cases for which the results of Grosshans [2] are applicable. Let R be the root system relative to a maximal torus T of G, and let D be a basis of R (which we identify with the Dynkin diagram of R). For a closed subset S of the set R + of positive roots, we denote by Us the corresponding standard unipotent subgroup of G, that is, U, = naES U,, where the product is taken in any fixed order and the U, are the root groups. For a subset I of D we denote S1 = R ‘\Z’I (so that U,, is the unipotent radical of a parabilic subgroup). Denote by C[G] the algebra of polynomial functions on G. The main result of this paper is the following.