148 research outputs found
Erratum to “Lefschetz Theory for Exterior Algebras and Fermionic Diagonal Coinvariants”
No full textThis erratum corrects the proof of the main result [1, Thm. 5.2] of [1, Sec. 5]. While this result is correct as stated, its proof is flawed
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Schur Positivity and Graded Ehrhart Theory
No full textChern plethysm, introduced by Billey, Rhoades, and Tewari, is a geometric way to produce Schur positive symmetric polynomials. In Chapter 1, we present combinatorial interpretations for the Schur expansions of special cases of Chern plethysm. We also exhibit a symmetric group module whose Frobenius characteristic is a symmetric function analog of one of these cases, generalizing a result of Reiner and Webb. In Chapter 2, we transition to a different geometric context: Ehrhart theory. First, we fully understand the orbit-harmonics quotients which arise from hypersimplices. We then interpret those graded vector spaces combinatorially in terms of tableaux, and prove one case of a vast conjecture of Reiner and Rhoades: that the harmonic algebra of a certain class of hypersimplices is finitely generated
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Webs for Flamingo Specht Modules
No full textA web basis of a representation of is a basis of the representation for which the action of can be understood through combinatorial rules called skein relations. In this thesis, we study web bases for two families of irreducible representations, indexed by the partitions and . The first was introduced by Rhoades and is indexed by noncrossing set partitions of . We use it to give a model for the top degree component of the fermionic diagonal coinvariant ring, and introduce another similar basis to model the entire fermionic diagonal coinvariant ring. We also give an embedding of the noncrossing set partition representation into an induction product of the Temperley-Lieb representation with a sign representation, thereby providing alternate proofs that the skein relations which define the noncrossing set partition representation are in fact well defined. The second web basis is new, and simultaneously generalizes the web basis of Kuperberg and the noncrossing set partition web basis. To define it and show it gives a basis, we draw on the combinatorics of Plabic graphs, jellyfish invariants, and weblike subgraphs
Hall-Littlewood polynomials and Hecke action on ordered set partitions
No full textWe construct an action of the Hecke algebra H-n(q) on a quotient of the polynomial ring F[x(1),..., x(n)], where F = Q(q). The dimension of our quotient ring is the number of k-block ordered set partitions of {1, 2,..., n}. This gives a quantum analog of a construction of Haglund-Rhoades-Shimozono and interpolates between their result at q = 1 and work of Huang-Rhoades at q = 0
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Rational Catalan Combinatorics
No full textGiven a finite Coxeter group W and a Coxeter element c, the W-noncrossing partitions are given by [1,c], the interval between 1 and c in W under the absolute order. When W is the symmetric group S_a, the noncrossing partitions turn out to be classical noncrossing partitions of [a] counted by the Catalan numbers. By attaching an additional integral paramenter b which is coprime to a, we define a set NC(a,b) of rational noncrossing partitions, a subset of the ordinary noncrossing partitions of [b-1]. We study the poset structure this set inherits from the poset of classical noncrossing partitions, ordered by refinement. We prove that NC(a,b) is closed under a dihedral action and that the rotation action on NC(a,b) exhibits the cyclic sieving phenomenon. We also generalize noncrossing parking functions to the rational setting and provide a character formula for the action of S_a X Z_{b-1} on a,b-noncrossing parking functions. Finally, we give a group-theoretic interpretation in type A for NC(a,b) in terms of compatible sequences
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Generalizations of the Coinvariant Algebra
No full textThe classical coinvariant algebra is the quotient of the polynomial ring in n variables by the ideal generated by polynomials that are invariant under the action of variable permutation. The classical coinvariant algebra is a fundamental object of study in the theory of algebraic combinatorics and a variety of generalizations of it have been defined. In this dissertation we will explore a variety of generalizations and refinements of the coinvariant algebra
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Tableaux formulas for Lascoux polynomials
No full textLascoux polynomials simultaneously generalize two famous families of polynomialsarising from geometry and representation theory: They are non-symmetric analogs
of Grassmannian stable Grothendieck polynomials, which represent Schubert classes
in the connective K-theory of Grassmannians. Additionally, they serve as non-
homogeneous analogs of key polynomials, the characters of Demazure modules. Both
of these families have classical combinatorial formulas involving tableaux. We further
generalize several of these formulas by establishing two combinatorial formulas for
Lascoux polynomials
A proof of the fermionic theta coinvariant conjecture
No full textLet (x1, ... , xn, y1, ... , yn) be a list of 2n commuting variables, (theta 1, ... , theta n, xi 1, ... , xi n) be a list of 2n anticommuting variables, and C[xn, yn] circle times perpendicular to{On, 4n} be the algebra generated by these variables. D'Adderio, Iraci, and Vanden Wyngaerd introduced the Theta operators on the ring of symmetric functions and used them to conjecture a formula for the quadruply -graded en-isomorphism type of C[xn,yn] circle times perpendicular to{On, 4n}/I where I is the ideal generated by en-invariants with vanishing constant term. We prove their conjecture in the 'purely fermionic setting' obtained by setting the commuting variables xi, yi equal to zero.(c) 2023 Elsevier B.V. All rights reserved
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Combinatorics in the Rational Shuffle Theorem and the Delta Conjecture
No full textThe Classical Shuffle Conjecture proposed by Haglund, Haiman, Loehr, Remmel and Ulyanov gives a well-studied combinatorial expression for the bigraded Frobenius characteristic of Sn-module of the ring of diagonal harmonics, which has been proved by Carlsson and Mellit as the Shuffle Theorem, stating that a symmetric function expression ∇en equals a generating function of combinatorial objects called parking functions. The Rational Shuffle Theorem of the expression Q_m,n(−1)^n of Mellit and the Delta Conjecture of the expression D'_ek en proposed by Haglund, Remmel and Wilson are two natural generalizations of the Shuffle Theorem. The primary goal of this dissertation is to prove some special cases of the conjectures, and compute the Schur function expansions of the corresponding symmetric function expressions. We explore several symmetries in the combinatorics of the coefficients that arise in the Schur function expansion of Q_m,n(−1)^n in the Rational Shuffle Theorem. Especially, we study the hook-shaped Schur function coefficients, and the Schur function expansion of Q_m,n(−1)^n in the case where m or n equals 3. We give a combinatorial proof that the coefficient of s_lambda in the Delta expression D_e2 en has a non-negative expansion in terms of q,t-analogues. We propose a new valley version conjecture of the expression D'_ek D_hr en, and we give a proof of the valley version conjecture of D'_ek D_hr en when t or q equals 0. Our work lead to many new results about the combinatorial objects in the conjectures, such as the Mahonian distribution in extended ordered multiset partitions and the straightening action in parking functions
Cyclic Sieving, Promotion, and Representation Theory
No full textWe prove a collection of conjectures due to Abuzzahab-Korson-Li-Meyer, Reiner, and White regarding the cyclic sieving phenomenon as it applies to jeu-de-taquin promotion on rectangular tableaux. To do this, we use Kazhdan-Lusztig theory and a characterization of the dual canonical basis of due to Skandera. Afterwards, we extend our results to analyzing the fixed points of a dihedral action on rectangular tableaux generated by promotion and evacuation, suggesting a possible sieving phenomenon for dihedral groups. Finally, we give applications of this theory to cyclic sieving phenomena involving reduced words for the long elements of hyperoctohedral groups, handshake patterns, and noncrossing partitions
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