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Mike Zabrocki

Publications and source records attributed to Mike Zabrocki.

At least 19 recordsLinked to original sources
arXiv2026

Rook characters as symmetric functions

We give several characterizations of an inhomogeneous basis of the ring of symmetric functions whose evaluations are the character values of the irreducible representations of the rook monoid (symmetric inverse semigroup). Using Schur--Weyl duality, we show that the transition coefficients of this basis with the power symmetric basis are the characters of the propagating partition algebra (dual symmetric inverse monoid algebra). In addition, the structure coefficients of this basis are equal to the coefficients in the smash (or Heisenberg) product of Schur functions.

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arXiv2026

A memorial tribute: Adriano Garsia (1928--2024)

Adriano Mario Garsia was born in Tunis on August 20, 1928, to a Tunisian-Italian family. He lived on a farm there until the end of World War II, then moved to Rome. After finishing high school, he was sent to the United States to live with relatives in Woyming and eventually made his way to California, becoming a student of Charles Loewner at Stanford in the early 1950s. Following his Ph.D., Adriano held positions at MIT, the University of Minnesota, and Caltech before joining the nascent mathematics department at the University of California, San Diego, in 1966 where he spent the remainder of his career. He passed away in San Diego on October 6, 2024, at the age of 96.

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arXiv2026

When are Hopf algebras determined by integer sequences?

We study the category of graded Hopf algebras that are free noncommutative, cocommutative, graded and connected from the perspective of the sequences of dimensions of the graded pieces. We show that a Hopf algebra exists with a given sequence of graded dimensions if and only if the ``INVERTi'' transformation of the sequence is nonnegative. We give conditions on the sequences of graded dimensions for two Hopf algebras $H$ and $K$ in this category under which there exists a surjective homomorphism from $H$ to $K$. We also give conditions such that an isomorphic copy of $H$ occurs as a Hopf subalgebra of $K$.

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arXiv2026

A geometric and generating function approach to plethysm

Plethysm coefficients $\mathsf{a}_{μ[ν]}^λ$ are the structure coefficients of the plethysm of Schur functions $s_μ[s_ν] = \sum_λ \mathsf{a}_{μ[ν]}^λs_λ$. We study a bivariate generating function of plethysm coefficients when $λ$ has bounded length. We show that this generating function is rational. A key step is MacMahon's combinatory analysis. When the bound on the length is $2$ we give an explicit geometric algorithm to compute it using $q$-Ehrhart theory. We give evidence that the generating function is the quantum Ehrhart series of a union of half-open polytopes and show that it satisfies a reciprocity theorem reminiscent of Ehrhart reciprocity. Furthermore, we give a set of linear recursions that completely describe the $\mathrm{SL}_2$-plethysm coefficients.

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arXiv2024

From quasi-symmetric to Schur expansions with applications to symmetric chain decompositions and plethysm

It is an important problem in algebraic combinatorics to deduce the Schur function expansion of a symmetric function whose expansion in terms of the fundamental quasisymmetric function is known. For example, formulas are known for the fundamental expansion of a Macdonald symmetric function and for the plethysm of two Schur functions, while the Schur expansions of these expressions are still elusive. Egge, Loehr and Warrington provided a method to obtain the Schur expansion from the fundamental expansion by replacing each quasisymmetric function by a Schur function (not necessarily indexed by a partition) and using straightening rules to obtain the Schur expansion. Here we provide a new method that only involves the coefficients of the quasisymmetric functions indexed by partitions and the quasi-Kostka matrix. As an application, we identify the lexicographically largest term in the Schur expansion of the plethysm of two Schur functions. We provide the Schur expansion of $s_w[s_h](x,y)$ for $w=2,3,4$ using novel symmetric chain decompositions of Young's lattice for partitions in a $w\times h$ box. For $w=4$, this is first known combinatorial expression for the coefficient of $s_λ$ in $s_{w}[s_{h}]$ for two-row partitions $λ$, and for $w=3$ the combinatorial expression is new.

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arXiv2024

The lattice of submonoids of the uniform block permutations containing the symmetric group

We study the lattice of submonoids of the uniform block permutation monoid containing the symmetric group (which is its group of units). We prove that this lattice is distributive under union and intersection by relating the submonoids containing the symmetric group to downsets in a new partial order on integer partitions. Furthermore, we show that the sizes of the $\mathscr{J}$-classes of the uniform block permutation monoid are sums of squares of dimensions of irreducible modules of the monoid algebra.

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arXiv2023

Representations of the quasi-partition algebras

The quasi-partition algebras were introduced by Daugherty and the first author as centralizers of the symmetric group. In this article, we give a more general definition of these algebras and give a construction of their simple modules. In addition, we introduce two new algebras, we give linear bases and show that for specializations of their parameters, these new algebras are isomorphic to centralizer algebras. We provide a generalized Bratteli diagram that illustrates how the representation theory of the three algebras discussed in this paper are related. Moreover, we give combinatorial formulas for the dimensions of the simple modules of these algebras.

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arXiv2022

Quasisymmetric harmonics of the exterior algebra

We study the ring of quasisymmetric polynomials in $n$ anticommuting (fermionic) variables. Let $R_n$ denote the polynomials in $n$ anticommuting variables. The main results of this paper show the following interesting facts about quasisymmetric polynomials in anticommuting variables: (1) The quasisymmetric polynomials in $R_n$ form a commutative sub-algebra of $R_n$. (2) There is a basis of the quotient of $R_n$ by the ideal $I_n$ generated by the quasisymmetric polynomials in $R_n$ that is indexed by ballot sequences. The Hilbert series of the quotient is given by $$ \text{Hilb}_{R_n/I_n}(q) = \sum_{k=0}^{\lfloor{n/2}\rfloor} f^{(n-k,k)} q^k\,,$$ where $f^{(n-k,k)}$ is the number of standard tableaux of shape $(n-k,k)$. (3) There is a basis of the ideal generated by quasisymmetric polynomials that is indexed by sequences that break the ballot condition

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arXiv2022

Plethysm and the algebra of uniform block permutations

We study the representation theory of the uniform block permutation algebra in the context of the representation theory of factorizable inverse monoids. The uniform block permutation algebra is a subalgebra of the partition algebra and is also known as the party algebra. We compute its characters and provide a Frobenius characteristic map to symmetric functions. This reveals connections of the characters of the uniform block permutation algebra and plethysms of Schur functions.

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arXiv2022

The Hopf structure of symmetric group characters as symmetric functions

In arXiv:1605.06672 the authors introduced inhomogeneous bases of the ring of symmetric functions. The elements in these bases have the property that they evaluate to characters of symmetric groups. In this article we develop further properties of these bases by proving product and coproduct formulae. In addition, we give the transition coefficients between the elementary symmetric functions and the irreducible character basis.

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arXiv2022

A basis for the Diagonal Harmonic Alternants

It will be shown here that there are differential operators $E,F$ and $H=[E,F]$ for each $n\ge 1$, acting on Diagonal Harmonics, yielding that $DH_n$ is a representation of $sl[2]$ (see [3] Chapter 3). Our main effort here is to use $sl[2]$ theory to predict a basis for the Diagonal Harmonic Alternants, $DHA_n$. It can be shown that the irreducible representations $sl[2]$ are all of the form $P,EP,E^2P,\cdots,E^kP$, with $FP=0$ and $E^{k+1}P=0$. The polynomial $P$ is known to be called a "String Starter". From $sl[2]$ theory it follows that $DHA_n$ is a direct sum of strings. Our main result so far is a formula for the number of string starters. A recent paper by Carlsson and Oblomkov (see [2]) constructs a basis for the space of Diagonal Coinvariants by Algebraic Geometrical tools. It would be interesting to see if any our results can be derived from theirs.

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arXiv2021

Symmetric group characters as symmetric functions

We introduce a basis of the symmetric functions that evaluates to the (irreducible) characters of the symmetric group, just as the Schur functions evaluate to the irreducible characters of $GL_n$ modules. Our main result gives three different characterizations for this basis. One of the characterizations shows that the structure coefficients for the (outer) product of these functions are the stable Kronecker coefficients. The results in this paper focus on developing the fundamental properties of this basis.

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arXiv2020

Howe duality of the symmetric group and a multiset partition algebra

We introduce the multiset partition algebra, ${\rm M\!P}_{r,k}(x)$, that has bases elements indexed by multiset partitions, where $x$ is an indeterminate and $r$ and $k$ are non-negative integers. This algebra can be realized as a diagram algebra that generalizes the partition algebra. When $x$ is an integer greater or equal to $2r$, we show that ${\rm M\!P}_{r,k}(x)$ is isomorphic to a centralizer algebra of the symmetric group, $S_n$, acting on the polynomial ring on the variables $x_{ij}$, $1\leq i \leq n$ and $1\leq j\leq k$. We describe the representations of ${\rm M\!P}_{r,k}(x)$, branching rule and restriction of its representations in the case that $x$ is an integer greater or equal to $2r$.

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arXiv2020

A combinatorial model for the decomposition of multivariate polynomial rings as $S_n$-modules

We consider the symmetric group $S_n$-module of the polynomial ring with $m$ sets of $n$ commuting variables and $m'$ sets of $n$ anti-commuting variables and show that the multiplicity of an irreducible indexed by the partition $λ$ (a partition of $n$) is the number of multiset tableaux of shape $λ$ satisfying certain column and row strict conditions. We also present a finite generating set for the ring of $S_n$ invariant polynomials of this ring.

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arXiv2020

An insertion algorithm on multiset partitions with applications to diagram algebras

We generalize the Robinson-Schensted-Knuth algorithm to the insertion of two row arrays of multisets. This generalization leads to new enumerative results that have representation theoretic interpretations as decompositions of centralizer algebras and the spaces they act on. In addition, restrictions on the multisets lead to further identities and representation theory analogues. For instance, we obtain a bijection between words of length $k$ with entries in $[n]$ and pairs of tableaux of the same shape with one being a standard Young tableau of size $n$ and the other being a standard multiset tableau of content $[k]$. We also obtain an algorithm from partition diagrams to pairs of a standard tableau and a standard multiset tableau of the same shape, which has the remarkable property that it is well-behaved with respect to restricting a representation to a subalgebra. This insertion algorithm matches recent representation-theoretic results of Halverson and Jacobson.

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arXiv2019

A module for the Delta conjecture

We define a module that is an extension of the diagonal harmonics and whose graded Frobenius characteristic is conjectured to be the symmetric function expression which appears in `the Delta conjecture' of Haglund, Remmel and Wilson [arXiv:1509.07058].

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arXiv2018

A proof of the $4$-variable Catalan polynomial of the Delta conjecture

In The Delta Conjecture (arxiv:1509.07058), Haglund, Remmel and Wilson introduced a four variable $q,t,z,w$ Catalan polynomial, so named because the specialization of this polynomial at the values $(q,t,z,w) = (1,1,0,0)$ is equal to the Catalan number $\frac{1}{n+1}\binom{2n}{n}$. We prove the compositional version of this conjecture (which implies the non-compositional version) that states that the coefficient of $s_{r,1^{n-r}}$ in the expression $Δ_{h_\ell} \nabla C_α$ is equal to a weighted sum over decorated Dyck paths.

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