On the depth of separating invariants for finite groups

Abstract. Consider a finite group G acting on a vector space V over a field k of characteristic p > 0. A separating algebra is a subalgebra A of the ring of invariants k[V]^G with the same point separation properties. In this article we compare the depth of an arbitrary separating algebra with that of the corresponding ring of invariants. We show that, in some special cases, the depth of A is bounded above by the depth of k[V]^G

Associated primes for cohomology modules

Let G be a finite group of order divisible by a prime p, and V a representation of G over a field of characteristic p. Let R denote the symmetric algebra on V*. We show that the associated primes of the R^G modules H^i(G,R) are the radicals of certain transfer ideals

Depth and detection in modular invariant theory

Let G be a finite group acting linearly on a vector space V over a field of characteristic p dividing the group order, and let R denote S(V∗). We study the R^G modules H^i(G, R), for i ≥ 0 with R^G itself as a special case. There are lower bounds for depth of (H^i(G, R)) and for depth(R^G). We show that a certain sufficient condition for their attainment (due to Fleischmann, Kemper and Shank) may be modified to give a condition which is both necessary and sufficient. We apply our main result to classify the representations of the Klein four-group for which depth(R^G) attains its lower bound. We also use our new condition to show that the if G = P × Q, with P a p-group and Q an abelian p'-group, then the depth of R G attains its lower bound if and only if the depth of R^P does so

The relative Heller operator and relative cohomology for the Klein 4-group

Let G be the Klein Four-group and let k be an arbitrary field of characteristic 2. A classification of indecomposable kG-modules is known. We calculate the relative cohomology groups H^i_χ(G,N) for every indecomposable kG-module N , where χ is the set of proper subgroups in G. This extends work of Pamuk and Yalcin to cohomology with non-trivial coefficients. We also show that all cup products in strictly positive degree in H^*_χ (G, k) are trivial

The separating variety for 2x2 matrix invariants

We study the action of the group G = GL_2(C) of invertible matrices over the complex numbers on the complex vector space V of n-tuples of 2x2 matrices. The algebra of invariants C[V]^G for this action is well-known, and has dimension 4n-3 and minimum generating set E_n with cardinality 1/6(n^3+11n). In recent work, Kaygorodov, Lopatin and Popov showed that this generating set is also a minimal separating set by inclusion, i.e. no proper subset is a separating set. This does not mean it has smallest possible cardinality among all separating sets. We show that if S is a separating set for C[V]^G then |S| is at least 5n-5. In particular for n=3, the set E_n is indeed of minimal cardinality, but for n>3 may not be so. We then show that a smaller separating set does in fact exist for n>4, We also prove similar results for the left-right action of SL_2(C)xSL_2(C) on V

Modular covariants of cyclic groups of order p

Let G be a cyclic group of order p and let V, W be kG-modules. We study the modules of covariants k[V,W]^G = (S(V^∗) ⊗ W)^G . Recall that G has exactly p inequivalent indecomposable kG-modules, denoted V_n (n = 1, . . . , p) and V_n has dimension n. For any n, we show that k[V_2,V_n]^G is a free k[V_2]^G- module (recovering a result of Broer and Chuai) and we give an explicit set of covariants generating k[V_2,V_n]^G freely over k[V_2]^G . For any n, we show that k[V_3,V_n]^G is a Cohen-Macaulay k[V_3]^G -module (again recovering a result of Broer and Chuai) and we give an explicit set of covariants which generate k[V 3 , V n ] G freely over a homogeneous system of parameters for k[V_3]^G . We also use our results to compute a minimal generating set for the transfer ideal of k[V_3]^G over a homogeneous system of parameters

Symmetric powers and modular invariants of elementary abelian p-groups

Let E be a elementary abelian p-group of order q = p^n. Let W be a faithful indecomposable representation of E with dimension 2 over a field k of characteristic p, and let V = S^m(W ) with m < q. We prove that the rings of invariants k[V ]^E are generated by elements of degree ≤ q and relative transfers. This extends recent work of Wehlau on modular invariants of cyclic groups of order p. If m < p we prove that k[V ]^E is generated by invariants of degree ≤ 2q −3, extending a result of Fleischmann, Sezer, Shank and Woodcock for cyclic groups of order p . Our methods are primarily representation-theoretic, and along the way we prove that for any d < q with d + m ≥ q, S^d (V^∗) is projective relative to the set of subgroups of E with order ≤ m, and that the sequence S^d (V^∗) is periodic with period q, modulo summands which are projective relative to the same set of subgroups. These results extend results of Almkvist and Fossum on cyclic groups of prime order

Field Guide to the Searobins (Prionotus and Bellator) in the Western North Atlantic

Species identifications of Prionotus and Bellator are often difficult under field conditions owing to the large number of species and their overlapping taxonomic characteristics. This key is intended to provide a simplified, accurate means to identify adult searobins greater than 10 cm standard length. All recognized species from the western North Atlantic, the Gulf of Mexico, and Caribbean Sea are included. (PDF file contains 30 pages.

Relative Factor Price Changes and Equity Prices

This paper suggests that the decline in equity prices, and thus in Tobin's average q, during the 1970s may be attributable to changes in expected relative factor prices. More specifically, q is shown to be a negative function of the extent to which current relative factor price expectations differ from those when capital was put in place. Because relative factor prices became more volatile after 1967, the observed decline in average q, and thus in stock prices, can be explained by the "relative price" hypothesis.