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Sophie Kriz

Publications and source records attributed to Sophie Kriz.

8 recordsLinked to original sources
arXiv2026

Howe duality over finite fields I: The two stable ranges

This is the first in a series of papers on type I Howe duality for finite fields, concerning the restriction of an oscillator representation of the symplectic group to a product of a symplectic and an orthogonal group. The goal of the series is describing this restriction completely explicitly. Applications (described in the third paper of the series) include demonstrating that the tensor pairs previously calculated by S.-Y. Pan as occuring with non-zero multiplicity occur with multiplicity 1, proving the type C case of the Gurevich-Howe rank conjecture, and giving a recursive formula for the characters of cuspidal unipotent representations. In this first paper, we construct the correspondence in the two so called stable ranges, where the rank of one of the factors is large enough with respect to the other.

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arXiv2026

Howe duality over finite fields II: Explicit stable computation

In this second paper of a series dedicated to type I Howe duality for finite fields, we explicitly describe the eta and zeta correspondences constructed in the first paper in terms of G. Lusztig's parametrization of the irreducible characters of finite groups of Lie type in the two so-called stable ranges. This identifies the stable eta and zeta correspondences among the pairs of irreducible representations whose occurence with non-zero multiplicity in the type I Howe duality correspondence was proved by S.-Y. Pan.

math.RT↗
arXiv2026

Howe duality over finite fields III: Full computation and the Gurevich-Howe conjectures

In this third paper in a series on type I Howe duality for finite fields, we give a complete description of the restriction of the oscillator representation over a finite field to products of dual pairs of symplectic and orthogonal groups in all cases that occur. We also provide a dictionary with the notation of S.-Y. Pan, who identified which tensor products of irreducible representations occur with non-zero multiplicity. As an application, we give a recursive construction of all irreducible complex representations of finite symplectic and orthogonal groups and a recursive formula for the characters of unipotent cuspidal representations. We also give a proof of the Gurevich-Howe rank and exhaustion conjectures for type C groups.

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arXiv2022

Some remarks on Mackey functors

The purpose of this paper is mainly to record how certain homotopy-theoretical constructions on ordinary G-equivariant cohomology spectra HM for a Mackey functor M, in particular products and duality, can be described on chain level. We will also discuss certain facts about modules over the constant Green functor $\underline{\mathbb{Z}}$.

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arXiv2021

Actads

In this paper, I introduce a new generalization of the concept of an operad, further generalizing the concept of an opetope introduced by Baez and Dolan, who used this for the definition of their version of non-strict $n$-categories. Opetopes arise from iterating a certain construction on operads called the $+$-construction, starting with monoids. The first step gives rise to plain operads, i.e. operads without symmetries. The permutation axiom in a symmetric operad, however, is an additional structure resulting from permutation of variables, independent of the structure of a monoid. Even though we can apply the $+$-construction to symmetric operads, there is the possibility of introducing a completely different kind of permutations on the higher levels by again permuting variables without regard to the structure on the previous levels. Defining and investigating these structures is the main purpose of this paper. The structures obtained in this way is what I call $n$-actads. In $n$-actads with $n>1$, the permutations on the different levels give rise to a certain special kind of $n$-fold category. I also explore the concept of iterated algebras over an $n$-actad (generalizing an algebra and module over an operad), and various types of iterated units. I give some examples of algebras over $2$-actads, and show how they can be used to construct certain new interesting homotopy types of operads. I also discuss a connection between actads and ordinal notation.

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arXiv2021

Equivariant cohomology and the super reciprocal plane of a hyperplane arrangement

In this paper, we investigate certain graded-commutative rings which are related to the reciprocal plane compactification of the coordinate ring of a complement of a hyperplane arrangement. We give a presentation of these rings by generators and defining relations. This presentation was used by Holler and I. Kriz to calculate the $\mathbb{Z}$-graded coefficients of localizations of ordinary $RO((\mathbb{Z}/p)^n)$-graded equivariant cohomology at a given set of representation spheres, and also more recently by the author in a generalization to the case of an arbitrary finite group. We also give an interpretation of these rings in terms of superschemes, which can be used to further illuminate their structure.

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arXiv2020

Notes on equivariant homology with constant coefficients

In this paper, for a finite group, we discuss a method for calculating equivariant homology with constant coefficients. We apply it to completely calculate the geometric fixed points of the equivariant spectrum representing equivariant (co)homology with constant coefficients. We also treat a more complicated example of inverting the standard representation in the equivariant homology of split extraspecial groups at the prime 2.

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