R. James Shank

Noether numbers for subrepresentations of cyclic groups of prime order

Let W be a finite-dimensional Z/p-module over a field, k, of characteristic p. The maximum degree of an indecomposable element of the algebra of invariants, k[W](Z/P), is called the Noether number of the representation, and is denoted by beta(W). A lower bound for beta(W) is derived, and it is shown that if U is a Z/p submodule of W, then beta(U) less than or equal to beta(W). A set of generators, in fact a SAGBI basis, is constructed for k[V2 circle plus V-3](Z/P), where V-n is the indecomposable Z/p-module of dimension n.

Computing Modular Invariants of p-groups

Let V be a finite dimensional representation of a p -group, G, over a field,k , of characteristic p. We show that there exists a choice of basis and monomial order for which the ring of invariants, kV ]G, has a finite SAGBI basis. We describe two algorithms for constructing a generating set for kV ] G. We use these methods to analyse k 2V3 ]U3where U3is the p -Sylow subgroup ofGL3 (Fp) and 2 V3is the sum of two copies of the canonical representation. We give a generating set for k 2 V3]U3forp= 3 and prove that the invariants fail to be Cohen?Macaulay forp 2. We also give a minimal generating set for kmV2 ]Z/pwere V2is the two-dimensional indecomposable representation of the cyclic group Z/p.

The transfer in modular invariant theory

Abstract We study the transfer homomorphism in modular invariant theory paying particular attention to the image of the transfer which is a proper non-zero ideal in the ring of invariants. We prove that, for a p -group over F p whose ring of invariants is a polynomial algebra, the image of the transfer is a principal ideal. We compute the image of the transfer for SL n ( F q ) and GL n ( F q ) showing that both ideals are principal. We prove that, for a permutation group, the image of the transfer is a radical ideal and for a cyclic permutation group the image of the transfer is a prime ideal.

Depth and Cohomological Connectivity in Modular Invariant Theory

the cohomological connectivity of the symmetric algebra S(V ) to be the smallest positive integer m such that H m (G;S(V ))6 0. We show that min dimK(V P ) +m + 1; dimK(V ) is a lower bound for the depth of S(V ) G . We characterize those representations for which the lower bound is sharp and give several examples of representations satisfying the criterion. In particular, we show that if G is p-nilpotent and P is cyclic then, for any modular representation, the depth of S(V ) G is min dimK(V P ) + 2; dimK(V ) .

Decomposing symmetric powers of certain modular representations of cyclic groups

For a prime number p, we construct a generating set for the ring of invariants for the p+1 dimensional indecomposable modular representation of a cyclic group of order p 2, and show that the Noether number for the representation is p 2 + p−3. We then use the constructed invariants to explicitly describe the decomposition of the symmetric algebra as a module over the group ring, confirming the Periodicity Conjecture of Ian Hughes and Gregor Kemper for this case. In the final section, we use our results to compute the Hilbert series for the corresponding ring of invariants together with some other related generating functions.

Lannes' T functor on summands of H*(B(Z/p)s)

Let H be the mod-p cohomology of the classifying space B(Z/p) thought of as an object in the category, U, of unstable modules over the Steenrod algebra. Lannes constructed a functor T:U→U which is left adjoint to the functor A→A⊗H. In this paper we evaluate T on the indecomposable U-summands of H ○ × s , the tensor product of s copies of H. Our formula involves the composition factors of certain tensor products of irreducible representations of the semigroup ring F p [M s,s (Z/p)]

Steenrod Algebra Module Maps from H ∗ (B( Z /p) n ) to H ∗ (B( Z /p) s )

Let Hon denote the mod-p cohomology of the classifying space B(Z/p)n as a module over the Steenrod algebra v . Adams, Gunawardena, and Miller have shown that the n x s matrices with entries in Z/p give a basis for the space of maps Homs (Ho?n, H?s) . For n and s relatively prime, we give a new basis for this space of maps using recent results of Campbell and Selick. The main advantage of this new basis is its compatibility with Campbell and Selicks direct sum decomposition of Hon into (pn _ 1) X-modules. Our applications are at the prime two. We describe the unique map from H to D(n) , the algebra of Dickson invariants in Hon , and we give the dimensions of the space of maps between the indecomposable summands of H?3.

Rings of invariants for the three-dimensional modular representations of elementary abelian p-groups of rank four

We show that the rings of invariants for the three-dimensional modular representations of an elementary abelian p-group of rank four are complete intersections with embedding dimension at most five. Our results confirm the conjectures of Campbell, Shank and Wehlau (Transform. Groups 18 (2013), 1‐22) for these representations.