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We use the gluing construction introduced by Jia Huang to explore the rings of invariants for a range of modular representations. We construct generating sets for the rings of invariants of the maximal parabolic subgroups of a finite symplectic group and their common Sylow $p$-subgroup. We also investigate the invariants of singular finite classical groups. We introduce parabolic gluing and use this construction to compute the invariant field of fractions for a range of representations. We use thin gluing to construct faithful representations of semidirect products and to determine the minimum dimension of a faithful representation of the semidirect product of a cyclic $p$-group acting on an elementary abelian $p$-group., Comment: Example 5.12 has been corrected and expanded

For a prime p > 2 and q = p n , we compute a finite generating set for the S L 2 ( F q ) -invariants of the second symmetric power representation, showing the invariants are a hypersurface and the field of fractions is a purely transcendental extension of the coefficient field. As an intermediate result, we show the invariants of the Sylow p -subgroups are also hypersurfaces.

For a prime p > 3 , we compute a finite generating set for the SL 2 ( F p ) -invariants of the third symmetric power representation. The proof relies on the construction of an infinite SAGBI basis and uses the Hilbert series calculation of Hughes and Kemper.

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., Comment: The new version incorporates minor corrections and an extended introduction

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 ) .

We study the cohomology modules H i (G;R) of a p-group G acting on a ring R of characteristic p, for i > 0. In particular, we are interested in the Cohen-Macaulay property and the depth of H i (G;R) regarded as an R G -module. We rst determine the support of H i (G;R), which turns out to be independent of i. Then we study the Cohen-Macaulay property for H 1 (G;R). Further results are restricted to the special case that G is cyclic and R is the symmetric algebra of a vector space on which G acts. We determine the depth of H i (G;R) for i odd and obtain results in certain cases for i even. Along the way, we determine the degrees in which the transfer map Tr G :R! R G has non-zero image.

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.

Let H H be the mod - p \bmod \text {-}p cohomology of the classifying space B ( Z / p ) B({\mathbf {Z}}/p) thought of as an object in the category, U \mathcal {U} , of unstable modules over the Steenrod algebra. Lannes constructed a functor T : U → U T:\mathcal {U} \to \mathcal {U} which is left adjoint to the functor A ↦ A ⊗ H A \mapsto A \otimes H . In this paper we evaluate T T on the indecomposable U \mathcal {U} -summands of H ⊗ s {H^{ \otimes s}} , the tensor product of s s copies of H 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 ) ] {{\mathbf {F}}_p}[{{\mathbf {M}}_{s,}}_s({\mathbf {Z}}/p)] . The main application is to determine the homotopy type of the space of maps from B ( Z / p ) ] B({\mathbf {Z}}/p)] to X X when X X is a wedge summand of the space Σ ( B ( Z / p ) s ) \Sigma (B{({\mathbf {Z}}/p)^s}) .

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.

Let H ⊗ n {H^{ \otimes n}} denote the mod − p \operatorname {mod}-p cohomology of the classifying space B ( Z / p ) n B{({\mathbf {Z}}/p)^n} as a module over the Steenrod algebra A \mathcal {A} . Adams, Gunawardena, and Miller have shown that the n × s n \times s matrices with entries in Z / p {\mathbf {Z}}/p give a basis for the space of maps Ho m A ( H ⊗ n , H ⊗ s ) {\text {Ho}}{{\text {m}}_\mathcal {A}}({H^{ \otimes n}},{H^{ \otimes s}}) . For n n and s 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 Selick’s direct sum decomposition of H ⊗ n {H^{ \otimes n}} into ( p n − 1 ) ({p^n} - 1) A \mathcal {A} -modules. Our applications are at the prime two. We describe the unique map from H ¯ \bar H to D ( n ) D(n) , the algebra of Dickson invariants in H ⊗ n {H^{ \otimes n}} , and we give the dimensions of the space of maps between the indecomposable summands of H ⊗ 3 {H^{ \otimes 3}} .

We consider the ring of coinvariants for modular representations of cyclic groups of prime order. For all cases for which explicit generators for the ring of invariants are known, we give a reduced Gröbner basis for the Hilbert ideal and the corresponding monomial basis for the coinvariants. We also describe the decomposition of the coinvariants as a module over the group ring. For one family of representations, we are able to describe the coinvariants despite the fact that an explicit generating set for the invariants is not known. In all cases our results confirm the conjecture of Harm Derksen and Gregor Kemper on degree bounds for generators of the Hilbert ideal. As an incidental result, we identify the coefficients of the monomials appearing in the orbit product of a terminal variable for the three-dimensional indecomposable representation.

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)]

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 Selick's 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.

We study the depth of the ring of invariants of SL 2 ( F p ) acting on the n th symmetric power of the natural two-dimensional representation for n p . These symmetric power representations are the irreducible representations of SL 2 ( F p ) over F p . We prove that, when the greatest common divisor of p − 1 and n is less than or equal to 2, the depth of the ring of invariants is 3. We also prove that the depth is 3 for n = 3, p ≠ 7 and n = 4, p ≠ 5. However, for n = 3, p = 7 the depth is 4 and for n = 4, p = 5 the depth is 5. In these two exceptional cases, the ring of invariants is Cohen–Macaulay.

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.