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Register a new userRational and Quasi-Permutation Representations of Holomorph of Cyclic p-Groups
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For a finite group $G$, let $p(G)$ denote the minimal degree of a faithful permutation representation of $G$. The minimal degree of a faithful representation of $G$ by quasi-permutation matrices over the fields $mathbb{C}$ and $mathbb{Q}$ are denoted by $c(G)$ and $q(G)$ respectively. In general $c(G)leq q(G)leq p(G)$ and either inequality may be strict. In this paper, we study the representation theory of the group $G =$ Hol$(C_{p^{n}})$, which is the holomorph of a cyclic group of order $p^n$, $p$ a prime. This group is metacyclic when $p$ is odd and metabelian but not metacyclic when $p=2$ and $n geq 3$. We explicitly describe the set of all isomorphism types of irreducible representations of $G$ over the field of complex numbers $mathbb{C}$ as well as the isomorphism types over the field of rational numbers $mathbb{Q}$. We compute the Wedderburn decomposition of the rational group algebra of $G$. Using the descriptions of the irreducible representations of $G$ over $mathbb{C}$ and over $mathbb{Q}$, we show that $c(G) = q(G) = p(G) = p^n$ for any prime $p$. The proofs are often different for the case of $p$ odd and $p=2$.
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We study Koopman and quasi-regular representations corresponding to the action of arbitrary weakly branch group G on the boundary of a rooted tree T. One of the main results is that in the case of a quasi-invariant Bernoulli measure on the boundary of T the corresponding Koopman representation of G is irreducible (under some general conditions). We also show that quasi-regular representations of G corresponding to different orbits and Koopman representations corresponding to different Bernoulli measures on the boundary of T are pairwise disjoint. This gives two continual collections of pairwise disjoint irreducible representations of a weakly branch group. Another corollary of our results is triviality of the centralizer of G in various groups of transformations on the boundary of T.
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We classify the localizing tensor ideals of the integral stable module category for any finite group $G$. This results in a generic classification of $mathbb{Z}[G]$-lattices of finite and infinite rank and globalizes the modular case established in celebrated work of Benson, Iyengar, and Krause. Further consequences include a verification of the generalized telescope conjecture in this context, a tensor product formula for integral cohomological support, as well as a generalization of Quillens stratification theorem for group cohomology. Our proof makes use of novel descent techniques for stratification in tensor-triangular geometry that are of independent interest.
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