Searcharxiv⌕ Search

arXiv subjects

Brendon Rhoades

Publications and source records attributed to Brendon Rhoades.

At least 37 records · Page 2Linked to original sources
arXiv2024

Higher Specht bases for generalizations of the coinvariant ring

The classical coinvariant ring $R_n$ is defined as the quotient of a polynomial ring in $n$ variables by the positive-degree $S_n$-invariants. It has a known basis that respects the decomposition of $R_n$ into irreducible $S_n$-modules, consisting of the higher specht polynomials due to Ariki, Terasoma, and Yamada. We provide an extension of the higher Specht basis to the generalized coinvariant rings $R_{n,k}$. We also give a conjectured higher Specht basis for the Garsia-Procesi modules $R_μ$, and provide a proof of the conjecture in the case of two-row partition shapes $μ$. We then combine these results to give a higher Specht basis for an infinite subfamily of the modules $R_{n,k,μ}$ recently defined by Griffin, which are a common generalization of $R_{n,k}$ and $R_μ$.

math.CO↗
arXiv2023

The combinatorics of supertorus sheaf cohomology

Affine superspace $\mathbb{C}^{1 \mid n}$ has a single bosonic coordinate $z$ and $n$ fermionic coordinates $θ_1, \dots, θ_n$. Let $M$ be the supertorus obtained by quotienting $\mathbb{C}^{1 \mid n}$ by the abelian group generated by the maps $S: (z,θ_1, \dots, θ_n) \mapsto (z + 1, θ_1, \dots, θ_n)$ and $T: (z, θ_1, \dots, θ_n) \mapsto (z + t, θ_1 + α_1, \dots, θ_n + α_n)$ where $t \in \mathbb{C}$ has positive imaginary part and $α_1, \dots, α_n$ are independent fermionic parameters. We compute the zeroth and first cohomology groups of the structure sheaf $\mathcal{O}$ of $M$ as doubly graded $\mathfrak{S}_n$-modules, exhibiting an instance of Serre duality between these groups. We use skein relations and noncrossing matchings to give a combinatorial presentation of $H^0(M,\mathcal{O})$ in terms of generators and relations.

math.CO↗
arXiv2023

The Hilbert series of the superspace coinvariant ring

Let $Ω_n$ be the ring of polynomial-valued holomorphic differential forms on complex $n$-space, referred to in physics as the superspace ring of rank $n$. The symmetric group $\mathfrak{S}_n$ acts diagonally on $Ω_n$ by permuting commuting and anticommuting generators simultaneously. We let $SI_n \subseteq Ω_n$ be the ideal generated by $\mathfrak{S}_n$-invariants with vanishing constant term and study the quotient $SR_n = Ω_n / SI_n$ of superspace by this ideal. We calculate the doubly-graded Hilbert series of $SR_n$ and prove an `operator theorem' which characterizes the harmonic space $SH_n \subseteq Ω_n$ attached to $SR_n$ in terms of the Vandermonde determinant and certain differential operators. Our methods employ commutative algebra results which were used in the study of Hessenberg varieties. Our results prove conjectures of N. Bergeron, Li, Machacek, Sulzgruber, Swanson, Wallach, and Zabrocki.

math.CO↗
arXiv2023

Haglund's conjecture for multi-$t$ Macdonald polynomials

We provide new approaches to prove identities for the modified Macdonald polynomials via their LLT expansions. As an application, we prove a conjecture of Haglund concerning the multi-$t$-Macdonald polynomials of two rows.

math.CO↗
arXiv2022

Zonotopal algebras, orbit harmonics, and Donaldson-Thomas invariants of symmetric quivers

We apply the method of orbit harmonics to the set of break divisors and orientable divisors on graphs to obtain the central and external zonotopal algebras respectively. We then relate a construction of Efimov in the context of cohomological Hall algebras to the central zonotopal algebra of a graph $G_{Q,γ}$ constructed from a symmetric quiver $Q$ with enough loops and a dimension vector $γ$. This provides a concrete combinatorial perspective on the former work, allowing us to identify the quantum Donaldson-Thomas invariants as the Hilbert series of the space of $S_γ$-invariants of the Postnikov-Shapiro slim subgraph space attached to $G_{Q,γ}$. The connection with orbit harmonics in turn allows us to give a manifestly nonnegative combinatorial interpretation to numerical Donaldson-Thomas invariants as the number of $S_γ$-orbits under the permutation action on the set of break divisors on $G_{Q,γ}$. We conclude with several representation-theoretic consequences, whose combinatorial ramifications may be of independent interest.

math.CO↗
arXiv2022

A proof of the fermionic Theta coinvariant conjecture

Let $(x_1, \dots, x_n, y_1, \dots, y_n)$ be a list of $2n$ commuting variables, $(θ_1, \dots, θ_n, ξ_1, \dots, ξ_n)$ be a list of $2n$ anticommuting variables, and $\mathbb{C}[X_n, Y_n] \otimes \wedge \{Θ_n, Ξ_n\}$ be the algebra generated by these variables. D'Adderio, Iraci, and Vanden Wyngaerd introduced the {\em Theta operators} on the ring of symmetric functions and used them to conjecture a formula for the quadruply-graded $\mathfrak{S}_n$-isomorphism type of $\mathbb{C}[X_n,Y_n] \otimes \wedge \{Θ_n, Ξ_n\}/I$ where $I$ is the ideal generated by $\mathfrak{S}_n$-invariants with vanishing constant term. We prove their conjecture in the `purely fermionic setting' obtained by setting the commuting variables equal $x_i, y_i$ equal to zero.

math.CO↗
arXiv2021

Set partitions, fermions, and skein relations

Let $Θ_n = (θ_1, \dots, θ_n)$ and $Ξ_n = (ξ_1, \dots, ξ_n)$ be two lists of $n$ variables and consider the diagonal action of $\mathfrak{S}_n$ on the exterior algebra $\wedge \{ Θ_n, Ξ_n \}$ generated by these variables. Jongwon Kim and the second author defined and studied the fermionic diagonal coinvariant ring $FDR_n$ obtained from $\wedge \{ Θ_n, Ξ_n \}$ by modding out by the $\mathfrak{S}_n$-invariants with vanishing constant term. On the other hand, the second author described an action of $\mathfrak{S}_n$ on the vector space with basis given by noncrossing set partitions of $\{1,\dots,n\}$ using a novel family of skein relations which resolve crossings in set partitions. We give an isomorphism between a natural Catalan-dimensional submodule of $FDR_n$ and the skein representation. To do this, we show that set partition skein relations arise naturally in the context of exterior algebras. Our approach yields an $\mathfrak{S}_n$-equivariant way to resolve crossings in set partitions. We use fermions to clarify, sharpen, and extend the theory of set partition crossing resolution.

math.CO↗
arXiv2021

Set superpartitions and superspace duality modules

The superspace ring $Ω_n$ is a rank $n$ polynomial ring tensor a rank $n$ exterior algebra. Using an extension of the Vandermonde determinant to $Ω_n$, the authors previously defined a family of doubly graded quotients $\mathbb{W}_{n,k}$ of $Ω_n$ which carry an action of the symmetric group $\mathfrak{S}_n$ and satisfy a bigraded version of Poincaré Duality. In this paper, we examine the duality modules $\mathbb{W}_{n,k}$ in greater detail. We describe a monomial basis of $\mathbb{W}_{n,k}$ and give combinatorial formulas for its bigraded Hilbert and Frobenius series. These formulas involve new combinatorial objects called {\em ordered superpartitions}. These are ordered set partitions $(B_1 \mid \cdots \mid B_k)$ of $\{1,\dots,n\}$ in which the non-minimal elements of any block $B_i$ may be barred or unbarred.

math.CO↗
arXiv2021

Packed Words and Quotient Rings

The coinvariant algebra is a quotient of the polynomial ring $\mathbb{Q}[x_1,\ldots,x_n]$ whose algebraic properties are governed by the combinatorics of permutations of length $n$. A word $w = w_1 \dots w_n$ over the positive integers is packed if whenever $i > 2$ appears as a letter of $w$, so does $i-1$. We introduce a quotient $S_n$ of $\mathbb{Q}[x_1,\ldots,x_n]$ which is governed by the combinatorics of packed words. We relate our quotient $S_n$ to the generalized coinvariant rings of Haglund, Rhoades, and Shimozono as well as the superspace coinvariant ring.

math.CO↗
arXiv2020

Cyclic sieving and orbit harmonics

Orbit harmonics is a tool in combinatorial representation theory which promotes the (ungraded) action of a linear group $G$ on a finite set $X$ to a graded action of $G$ on a polynomial ring quotient by viewing $X$ as a $G$-stable point locus in $\mathbb{C}^n$. The cyclic sieving phenomenon is a notion in enumerative combinatorics which encapsulates the fixed-point structure of the action of a finite cyclic group $C$ on a finite set $X$ in terms of root-of-unity evaluations of an auxiliary polynomial $X(q)$. We apply orbit harmonics to prove cyclic sieving results.

math.CO↗
arXiv2020

Increasing Subsequences and Kronecker Coefficients

It has been conjectured by W. Chen that the distribution of the length of the longest increasing subsequence in a uniformly random permutation is log-concave. We propose a stronger version of this conjecture which involves the Kronecker coefficients of the symmetric group.

math.CO↗
arXiv2020

Harmonic bases for generalized coinvariant algebras

Let $k \leq n$ be nonnegative integers and let $λ$ be a partition of $k$. S. Griffin recently introduced a quotient $R_{n,λ}$ of the polynomial ring $\mathbb{Q}[x_1, \dots, x_n]$ in $n$ variables which simultaneously generalizes the Delta Conjecture coinvariant rings of Haglund-Rhoades-Shimozono and the cohomology rings of Springer fibers studied by Tanisaki and Garsia-Procesi. We describe the space $V_{n,λ}$ of harmonics attached to $R_{n,λ}$ and produce a harmonic basis of $R_{n,λ}$ indexed by certain ordered set partitions $\mathcal{OP}_{n,λ}$. The combinatorics of this basis is governed by a new extension of the {\em Lehmer code} of a permutation to $\mathcal{OP}_{n, λ}$.

math.CO↗
arXiv2020

Lefschetz theory for exterior algebras and fermionic diagonal coinvariants

Let $W$ be an irreducible complex reflection group acting on its reflection representation $V$. We consider the doubly graded action of $W$ on the exterior algebra $\wedge (V \oplus V^*)$ as well as its quotient $DR_W := \wedge (V \oplus V^*)/ \langle \wedge (V \oplus V^*)^{W}_+ \rangle$ by the ideal generated by its homogeneous $W$-invariants with vanishing constant term. We describe the bigraded isomorphism type of $DR_W$; when $W = \mathfrak{S}_n$ is the symmetric group, the answer is a difference of Kronecker products of hook-shaped $\mathfrak{S}_n$-modules. We relate the Hilbert series of $DR_W$ to the (type A) Catalan and Narayana numbers and describe a standard monomial basis of $DR_W$ using a variant of Motzkin paths. Our methods are type-uniform and involve a Lefschetz-like theory which applies to the exterior algebra $\wedge (V \oplus V^*)$.

math.CO↗
arXiv2019

Spanning subspace configurations and representation stability

Let $V_1, V_2, V_3, \dots $ be a sequence of $\mathbb{Q}$-vector spaces where $V_n$ carries an action of $\mathfrak{S}_n$ for each $n$. {\em Representation stability} and {\em multiplicity stability} are two related notions of when the sequence $V_n$ has a limit. An important source of stability phenomena arises in the case where $V_n$ is the $d^{th}$ homology group (for fixed $d$) of the configuration space of $n$ distinct points in some fixed topological space $X$. We replace these configuration spaces with the variety $X_{n,k}$ of {\em spanning configurations} of $n$-tuples $(\ell_1, \dots, \ell_n)$ of lines in $\mathbb{C}^k$ which satisfy $\ell_1 + \cdots + \ell_n = \mathbb{C}^k$ as vector spaces. We study stability phenomena for the homology groups $H_d(X_{n,k})$ as the parameter $(n,k)$ grows.

math.CO↗
arXiv2019

Vandermondes in superspace

Superspace of rank $n$ is a $\mathbb{Q}$-algebra with $n$ commuting generators $x_1, \dots, x_n$ and $n$ anticommuting generators $θ_1, \dots, θ_n$. We present an extension of the Vandermonde determinant to superspace which depends on a sequence $\mathbf{a} = (a_1, \dots, a_r)$ of nonnegative integers of length $r \leq n$. We use superspace Vandermondes to construct graded representations of the symmetric group. This construction recovers hook-shaped Tanisaki quotients, the coinvariant ring for the Delta Conjecture constructed by Haglund, Rhoades, and Shimozono, and a superspace quotient related to positroids and Chern plethysm constructed by Billey, Rhoades, and Tewari. We define a notion of partial differentiation with respect to anticommuting variables to construct doubly graded modules from superspace Vandermondes. These doubly graded modules carry a natural ring structure which satisfies a 2-dimensional version of Poincaré duality. The application of polarization operators gives rise to other bigraded modules which give a conjectural module for the symmetric function $Δ'_{e_{k-1}} e_n$ appearing in the Delta Conjecture of Haglund, Remmel, and Wilson.

math.CO↗
arXiv2019

Line configurations and r-Stirling partitions

A set partition of $[n] := \{1, 2, \dots, n \}$ is called {\em $r$-Stirling} if the numbers $1, 2, \dots, r$ belong to distinct blocks. Haglund, Rhoades, and Shimozono constructed graded ring $R_{n,k}$ depending on two positive integers $k \leq n$ whose algebraic properties are governed by the combinatorics of ordered set partitions of $[n]$ with $k$ blocks. We introduce a variant $R_{n,k}^{(r)}$ of this quotient for ordered $r$-Stirling partitions which depends on three integers $r \leq k \leq n$. We describe the standard monomial basis of $R_{n,k}^{(r)}$ and use the combinatorial notion of the {\em coinversion code} of an ordered set partition to reprove and generalize some results of Haglund et.\ al.\ in a more direct way. Furthermore, we introduce a variety $X_{n,k}^{(r)}$ of line arrangements whose cohomology is presented as the integral form of $R_{n,k}^{(r)}$, generalizing results of Pawlowski and Rhoades.

math.CO↗