arXiv subjects

Franco Saliola

Publications and source records attributed to Franco Saliola.

At least 19 recordsLinked to original sources

arXiv2025

A geometric and generating function approach to plethysm

Plethysm coefficients $\mathsf{a}_{\mu[\nu]}^\lambda$ are the structure coefficients of the plethysm of Schur functions $s_\mu[s_\nu] = \sum_{\lambda} \mathsf{a}_{\mu[\nu]}^\lambda s_\lambda$. We study a bivariate generating function of plethysm coefficients when $\lambda$ 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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arXiv2025

Factorizations in Hecke algebras I: long cycle factorizations and Jucys-Murphy elements

Given a permutation, there is a well-developed literature studying the number of ways one can factor it into a product of other permutations subject to certain conditions. We initiate the analogous theory for the type A Iwahori-Hecke algebra by generalizing the notion of factorization in terms of the Jucys-Murphy elements. Some of the oldest and most foundational factorization results for the symmetric groups pertain to the long cycle. Our main results give q-deformations of these long cycle factorizations and reveal q-binomial, q-Catalan, and q-Narayana numbers along the way.

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arXiv2025

The $q$-deformed random-to-random family in the Hecke algebra

We generalize Reiner--Saliola--Welker's well-known but mysterious family of *$k$-random-to-random shuffles* from Markov chains on symmetric groups to Markov chains on the Type-$A$ Iwahori--Hecke algebras. We prove that the family of operators pairwise commutes and has eigenvalues that are polynomials in $q$ with non-negative integer coefficients. Our work generalizes work of Reiner--Saliola--Welker and Lafreni\`ere for the symmetric group, and simplifies all known proofs in this case.

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arXiv2024

A quantum Murnaghan--Nakayama rule for the flag manifold

In this paper, we give a rule for the multiplication of a Schubert class by a tautological class in the (small) quantum cohomology ring of the flag manifold. As an intermediate step, we establish a formula for the multiplication of a Schubert class by a quantum Schur polynomial indexed by a hook partition. This entails a detailed analysis of chains and intervals in the quantum Bruhat order. This analysis allows us to use results of Leung--Li and of Postnikov to reduce quantum products by hook Schur polynomials to the (known) classical product.

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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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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 $s_{\lambda}$ in $s_{w}[s_{h}]$ for two-row partitions $\lambda$, and for $w=3$ the combinatorial expression is new.

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arXiv2023

The Primitive Eulerian polynomial

We introduce the Primitive Eulerian polynomial $P_{\cal A}(z)$ of a central hyperplane arrangement ${\cal A}$. It is a reparametrization of its cocharacteristic polynomial. Previous work of the first author implicitly show that, for simplicial arrangements, $P_{\cal A}(z)$ has nonnegative coefficients. For reflection arrangements of type A and B, the same work interprets the coefficients of $P_{\cal A}(z)$ using the (flag)excedance statistic on (signed) permutations. The main result of this article is to provide an interpretation of the coefficients of $P_{\cal A}(z)$ for all simplicial arrangements only using the geometry and combinatorics of ${\cal A}$. This new interpretation sheds more light to the case of reflection arrangements and, for the first time, gives combinatorial meaning to the coefficients of the Primitive Eulerian polynomial of the reflection arrangement of type D. In type B, we find a connection between the Primitive Eulerian polynomial and the $1/2$-Eulerian polynomial of Savage and Viswanathan (2012). We present some real-rootedness results and conjectures for $P_{\cal A}(z)$.

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arXiv2023

Left Regular Bands of Groups and the Mantaci--Reutenauer algebra

We develop the idempotent theory for algebras over a class of semigroups called left regular bands of groups (LRBGs), which simultaneously generalize group algebras of finite groups and left regular band (LRB) algebras. Our techniques weave together the representation theory of finite groups and LRBs, opening the door for a systematic study of LRBGs in an analogous way to LRBs. We apply our results to construct complete systems of primitive orthogonal idempotents in the Mantaci--Reutenauer algebra ${\sf{MR}}_n[G]$ associated to any finite group $G$. When $G$ is abelian, we give closed form expressions for these idempotents, and when $G$ is the cyclic group of order two, we prove that these recover idempotents introduced by Vazirani.

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arXiv2023

Quivers of stylic algebras

We construct a complete system of primitive orthogonal idempotents and give an explicit quiver presentation of the monoid algebra of the stylic monoid introduced by Abram and Reutenauer [arXiv:2106.06556].

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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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arXiv2021

Stable set polytopes and their 1-skeleta

We characterize the edges of two classes of $0/1$-polytopes. The first class corresponds to the stable set polytope of a graph $G$ and includes chain polytopes of posets, some instances of matroid independence polytopes, as well as newly-defined polytopes whose vertices correspond to noncrossing set partitions. In analogy with matroid basis polytopes, the second class is obtained by considering the stable sets of maximal cardinality. We investigate how the class of $0/1$-polytopes whose edges satisfy our characterization is situated within the hierarchy of $0/1$-polytopes. This includes the class of matroid polytopes. We also study the diameter of these classes of polytopes and improve slightly on the Hirsch bound.

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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 positivity conjecture on the structure constants of shifted Jack functions

We consider Jack polynomials $J_\lambda$ and their shifted analogue $J^#_\lambda$. In 1989, Stanley conjectured that $\langle J_\mu J_\nu, J_\lambda \rangle$ is a polynomial with nonnegative coefficients in the parameter $\alpha$. In this note, we extend this conjecture to the case of shifted Jack polynomials.

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arXiv2018

Cell complexes, poset topology and the representation theory of algebras arising in algebraic combinatorics and discrete geometry

In recent years it has been noted that a number of combinatorial structures such as real and complex hyperplane arrangements, interval greedoids, matroids and oriented matroids have the structure of a finite monoid called a left regular band. Random walks on the monoid model a number of interesting Markov chains such as the Tsetlin library and riffle shuffle. The representation theory of left regular bands then comes into play and has had a major influence on both the combinatorics and the probability theory associated to such structures. In a recent paper, the authors established a close connection between algebraic and combinatorial invariants of a left regular band by showing that certain homological invariants of the algebra of a left regular band coincide with the cohomology of order complexes of posets naturally associated to the left regular band. The purpose of the present monograph is to further develop and deepen the connection between left regular bands and poset topology. This allows us to compute finite projective resolutions of all simple modules of left regular band algebras over fields and much more. In the process, we are led to define the class of CW left regular bands as the class of left regular bands whose associated posets are the face posets of regular CW complexes. Most of the examples that have arisen in the literature belong to this class. A new and important class of examples is a left regular band structure on the face poset of a CAT(0) cube complex. Also, the recently introduced notion of a COM (complex of oriented matroids or conditional oriented matroid) fits nicely into our setting and includes CAT(0) cube complexes and certain more general CAT(0) zonotopal complexes. A fairly complete picture of the representation theory for CW left regular bands is obtained.

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arXiv2017

Spectral analysis of random-to-random Markov chains

We compute the eigenvalues and eigenspaces of random-to-random Markov chains. We use a family of maps which reveal a remarkable recursive structure of the eigenspaces, yielding an explicit and effective construction of all eigenbases starting from bases of the kernels.

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arXiv2016

Multiplicative structures of the immaculate basis of non-commutative symmetric functions

We continue our development of a new basis for the algebra of non-commutative symmetric functions. This basis is analogous to the Schur basis for the algebra of symmetric functions, and it shares many of its wonderful properties. For instance, in this article we describe non-commutative versions of the Littlewood-Richardson rule and the Murnaghan-Nakayama rule. A surprising relation develops among non-commutative Littlewood-Richardson coefficients, which has implications to the commutative case. Finally, we interpret these new coefficients geometrically as the number of integer points inside a certain polytope.

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arXiv2013

Poset topology and homological invariants of algebras arising in algebraic combinatorics

We present a beautiful interplay between combinatorial topology and homological algebra for a class of monoids that arise naturally in algebraic combinatorics. We explore several applications of this interplay. For instance, we provide a new interpretation of the Leray number of a clique complex in terms of non-commutative algebra. Résumé. Nous présentons une magnifique interaction entre la topologie combinatoire et l'algèbre homologique d'une classe de monoïdes qui figurent naturellement dans la combinatoire algébrique. Nous explorons plusieurs applications de cette interaction. Par exemple, nous introduisons une nouvelle interprétation du nombre de Leray d'un complexe de clique en termes de la dimension globale d'une certaine algèbre non commutative.

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arXiv2013

A lift of the Schur and Hall-Littlewood bases to non-commutative symmetric functions

We introduce a new basis of the non-commutative symmetric functions whose commutative images are Schur functions. Dually, we build a basis of the quasi-symmetric functions which expand positively in the fundamental quasi-symmetric functions and decompose Schur functions. We then use the basis to construct a non-commutative lift of the Hall-Littlewood symmetric functions with similar properties to their commutative counterparts.

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