Browsing by Subject "Representation Theory"

Now showing 1 - 5 of 5
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    The Non-Split bessel Model on GSp(4) as an Iwahori-Hecke Algebra Module
    (2017-06) Grodzicki, William
    We realize the non-split Bessel model of Novodvorsky and Piatetski-Shapiro as a generalized Gelfand-Graev representation of GSp(4), as defined by Kawanaka. Our primary goal is to calculate the values of Iwahori-fixed vectors of unramified principal series representations in the Bessel model. On the path to achieving this goal, we will first use Mackey theory to realize the Bessel functional as an integral - as a result, we will reestablish the uniqueness and existence of a Bessel model for principal series representations, originally proved by Novodvorsky and Piatetski-Shapiro and by Bump, Friedberg, and Furusawa, respectively. Inspired by the work of Brubaker, Bump, and Friedberg, our method of calculation takes advantage of the connection between the Iwahori-fixed vectors in the Bessel model and a certain linear character of the Hecke algebra of GSp(4). We will also provide a detailed description of the conjectural program connecting characters of the Hecke algebra for a more general reductive group G with multiplicity-free models of principal series representations. In particular, we will focus on the role played by the Springer correspondence in this program. Additionally, using the formulas we develop for the Iwahori-fixed vectors, we provide an explicit alternator expression for the spherical vector in the Bessel model which matches previous results of Bump, Friedberg, and Furusawa.
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    Parametrizations of Irreducible Rational Representations of Coxeter Groups
    (2024-04) Corsi, Craig
    We develop two closely related methods for parametrizing the rational irreducible characters of an arbitrary finite Coxeter group W. The goal is to provide a uniform approach to such a parametrization, independent of Coxeter type. The two methods generalize two approaches to describing the irreducible representations of the symmetric groups, which coincide in type A but do not coincide generally. Our methods associate characters to pairs of reflection subgroups, in one case by considering common constituents of permutation and signed permutation modules, and in the other case by a generalization of the Specht modules. We ask whether, using either method, we can identify a set of subgroup pairs for which the matrix of multiplicities of rational irreducibles in the characters associated to these subgroup pairs is unitriangular. Such a unitriangular matrix provides a parametrization of the irreducible rational characters. For the noncrystallographic types H and I, we give a positive answer to this question. In type H, we show computationally that we can parametrize the irreducible rational characters of H4 using generalized Specht modules, and we can parametrize the irreducible characters of H3 using both methods. Moreover, we give an explicit decomposition of the generalized common constituents for the dihedral groups I2(n) for all n, and we prove that we can always exhibit a unitriangular multiplicity matrix using generalized common constituents. In type A our theory coincides with the classical theory of Specht modules and does not give any new information. In type B the approach we take is closely related to an existing parametrization of the irreducible characters, but it appears to have some novel elements.
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    The Representation Theory of Transporter Categories
    (2022-05) Coopergard, Ryan
    In this paper, we extend the work of Diveris, Purin and Webb [5] to explore the structureof Auslander-Reiten quiver of Db(kP ⋊ G) and kP ⋊ G where G is a finite group, P is a finite poset, and P ⋊ G is the resulting transporter category. In particular, we show that given a transporter category P ⋊ G, a portion of the Auslander-Reiten quiver of Db(k[a, b] ⋊ Gb), where [a, b] ⋊ Gb is a subcategory which meets certain conditions. is copied into the Auslander-Reiten quiver of Db(kP ⋊ G). Moreover, we define a class of transporter categories, ICT, for which we can construct a slice of a component of the Auslander-Reiten quiver of Db(kP ⋊ G). This allows us to classify the transporter categories in ICT of finite representation type. We conclude with a connection to Young’s lattice of partitions.
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    Symmetries of tensors
    (2009-09) Berget, Andrew Schaffer
    This thesis studies the symmetries of a fixed tensors by looking at certain group representations this tensor generates. We are particularly interested in the case that the tensor can be written as v 1 ⊗ · · · ⊗ v n , where the v i are selected from a complex vector space. The general linear group representation generated by such a tensor contains subtle information about the matroid M ( v ) of the vector configuration v 1 , · · ·, v n . To begin, we prove the basic results about representations of this form. We give two useful ways of describing these representations, one in terms of symmetric group representations, the other in terms of degeneracy loci over Grassmannians. Some of these results are equivalent to results that have appeared in the literature. When this is the case, we have given new, short proofs of the known results. We will prove that the multiplicities of hook shaped irreducibles in the representation generated by v 1 ⊗ · · · ⊗ v n are determined by the no broken circuit complex of M ( v ). To do this, we prove a much stronger result about the structure of vector subspace of Sym V spanned by the products Π i∈S v i , where S ranges over all subsets of [ n ]. The result states that this vector space has a direct sum decomposition that determines the Tutte polynomial of M ( v ). We will use a combinatorial basis of the vector space generated the products of the linear forms to completely describe the representation generated by a decomposable tensor when its matroid M ( v ) has rank two. Next we consider a representation of the symmetric group associated to every matroid. It is universal in the sense that if v 1 , . . . , v n is a realization of the matroid then the representation for the matroid provides non-trivial restrictions on the decomposition of the representation generated by the tensor product of the vectors. A complete combinatorial characterization of this representation is proven for parallel extensions of Schubert matroids. We also describe the multiplicity of hook shapes in this representation for all matroids. The contents of this thesis will always be freely available online in the most current version. Simply search for my name and the title of the thesis. Please do not ever pay for this document.
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    A Uniform Approach towards the Local Gan-Gross-Prasad Conjecture
    (2024-06) Chen, Cheng
    The local Gan-Gross-Prasad conjecture is a generalization of the branching problem toclassical groups over local fields of characteristic zero. The conjecture speculates on the multiplicity, that is, the dimension of the Bessel models and Fourier-Jacobi models in an irreducible admissible representation. Equivalent conditions for the multiplicity equaling to one is given in [GP92, GP94, GGP12]. J.-L. Waldspurger did the pioneer’s work and proved the Bessel special orthogonal cases for tempered parameters over non-Archimedean local fields in [Wald10, Wald12a, Wald12b, Wald12c]. C. Mœglin and Waldspurger proved that case for generic parameters in [MW12]. The proof for the conjecture is almost completed but the proof for some cases used a different philosophy. This thesis aims to generalize Mœglin and Waldspurger’s approach to formulate a relatively uniform proof for all cases.