Browsing by Subject "Algebraic Combinatorics"

Now showing 1 - 3 of 3
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    Generalizations of Cluster Algebras from Triangulated Surfaces
    (2021-12) Banaian, Esther
    Cluster algebras, first defined by Fomin and Zelevinsky have proven to be a fascinating mathematical object, with connections in a wide range of mathematical fields as well as in physics. Many cluster algebras with desirable properties arise from triangulated surfaces. The goal of this thesis is to explore various generalizations of the narrative of cluster algebras from surfaces and to see which nice properties continue to hold. One fruitful direction is working with generalized cluster algebras from triangulated orbifolds. We give a combinatorial proof of positivity for such generalized cluster algebras by generalizing the snake graphs to the orbifold setting. We also show that our snake graphs give good expansions for generalized arcs and closed curves in an orbifold by showing the expansion formula satisfies the skein relations. We then turn to some applications of this snake graph expansion formula. The first is a generalization of the Markov numbers. Markov numbers can be seen as cluster variables from a certain cluster algebra with each initial cluster variable set to 1. We take a natural generalization of this cluster algebra and again specialize the initial variables to 1 to produce this new family of numbers. Since this generalized cluster algebra comes from an orbifold, we can explicitly describe the shape of the snake graphs corresponding to these numbers. The second application of the snake graph expansion formula is algebraic. Caldero and Chapoton gave a map which takes a module over an algebra and produces a Laurent polynomial; in some cases, the image of this map will be in a cluster algebra related to the original algebra. Using the snake graph expansion formula, we study the result of this map on a class of algebras determined by certain orbifolds. We also look to a different generalization by studying friezes and frieze patterns on surfaces with dissections, following work by Holm and Jorgensen. Caldero and Chapoton showed a connection between frieze patterns and cluster algebras. We study the space of all frieze patterns from dissections and give an algorithm to determine when an arbitrary frieze pattern comes from a dissection. We also give a combinatorial interpretation of the entries.
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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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    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.