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Nicholas Mayers

Publications and source records attributed to Nicholas Mayers.

At least 19 recordsLinked to original sources
arXiv2026

Varieties of chain complexes and mixed dimer covers

A quiver representation consists of a collection of vector spaces along with a set of arrows, which are linear maps between these spaces. In this work, we study quiver representations in equioriented type $A$ which are also chain complexes; that is, in which consecutive arrows compose to zero. We show that orbits of these representations under a change of basis action are in bijection with mixed dimer covers of a $2 \times n$ grid graph. The latter object can be endowed with a partial order which is a distributive lattice, and we show that the degeneration order on the orbits of chain complexes is a coarsening of this partial order. In addition, we use recent matrix formulae of Claussen and Ovenhouse to enumerate these orbits. This also computes the Kostant partition function applied to height-restricted, type $A$ roots. When the dimension vector is uniform, we discuss a correspondence with paths of a beam of light bouncing between glass plates and give an explicit generating function.

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arXiv2026

The nilradical of a seaweed algebra

Seaweed subalgebras of $\mathfrak{gl}(n,\mathbb{C})$ and $\mathfrak{sl}(n,\mathbb{C})$ are combinatorially defined matrix Lie algebras whose index admits a closed-form description in terms of an associated graph called a meander. In this paper, we study the nilradicals of these algebras with our main result establishing an explicit formula for their index in terms of an edge-weighted variation of the meander. We further prove that each such nilradical decomposes as a direct sum of the center of the seaweed subalgebra with a nilpotent Lie poset algebra, and we provide a meander-theoretic procedure for recovering the underlying poset.

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arXiv2026

The quantum k-Bruhat order

In this paper, we extend the study of the quantum $k$-Bruhat order initiated in the work of Benedetti, Bergeron, Colmenarejo, Saliola, and Sottile concerning the quantum Murnaghan-Nakayama rule. Specifically, identifying maximal chains in intervals of the quantum $k$-Bruhat order with sequences of transpositions, we investigate a naturally associated free monoid $F_n^{\mathbf{q}}$ with an action on a $q$-extension of $S_n$, denoted $S_n[\mathbf{q}]$, which encodes the chain structure of the quantum $k$-Bruhat order. Aside from numerous structural results, our main contribution is an identification of a large family of equivalences satisfied by the elements of $F_n^{\mathbf{q}}$ as operators on $S_n[\mathbf{q}]$. In fact, we conjecture that our list of equivalences is complete. As a consequence of the quantum Monk's rule, a complete understanding of such equivalences can be used to gain information about the multiplicative structure of quantum Schubert polynomials.

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arXiv2025

Two combinatorial puzzles arising from the theory of Kohnert polynomials

Motivated by recent work of Hanser and Mayers, we study two combinatorial puzzles arising from the theory of Kohnert polynomials. Such polynomials are defined as generating polynomials for certain collections of diagrams consisting of unit cells arranged in the first quadrant generated from an initial "seed diagram" by applying what are called "Kohnert moves". Each Kohnert move affects the position of at most cell of a diagram, attempting to move the rightmost cell of a given row to the first available position below and in the same column. In this paper, we study the combinatorial puzzles defined as follows: given a diagram $D$, form a diagram that is fixed by all Kohnert moves by applying either the fewest or most possible number of Kohnert moves. For both puzzles, we find complete solutions as well as methods for combinatorially computing the associated number of Kohnert moves in terms of the initial diagram $D$.

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arXiv2025

Dimensions of toggleability spaces

We establish a conjecture of Defant, Hopkins, Poznanovi\'{c}, and Propp concerning the dimensions of toggleability spaces for products of chains, shifted staircases, type-A root posets, and type-B posets. Generalizing this result, we show that for a larger family of posets defined by restricted diagrams, the dimensions of toggleability spaces are equal to the rank of the poset plus one. As part of our approach, we build upon the technique of rook statistics introduced by Chan, Haddadan, Hopkins, and Moci.

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arXiv2025

Ghost Kohnert posets

Recently, Pan and Yu showed that Lascoux polynomials can be defined in terms of certain collections of diagrams consisting of unit cells arranged in the first quadrant. Starting from certain initial diagrams, one forms a finite set of diagrams by applying two types of moves: Kohnert and ghost moves. Both moves cause at most one cell to move to a lower row with ghost moves leaving a new "ghost cell" in its place. Each diagram formed in this way defines a monomial in the associated Lascoux polynomial. Restricting attention to diagrams formed by applying sequences of only Kohnert moves in the definition of Lascoux polynomials, one obtains the family of key polynomials. Recent articles have considered a poset structure on the collections of diagrams formed when one uses only Kohnert moves. In general, these posets are not "well-behaved," not usually having desirable poset properties. Here, as an intermediate step to studying the analogous posets associated with Lascoux polynomials, we consider the posets formed by restricting attention to those diagrams formed by using only ghost moves. Unlike in the case of Kohnert posets, we show that such "ghost Kohnert posets" are always ranked join semi-lattices. In addition, we establish a necessary condition for when ghost Kohnert posets are bounded and, consequently, lattices.

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arXiv2024

The support of Kostant's weight multiplicity formula is an order ideal in the weak Bruhat order

For integral weights $\lambda$ and $\mu$ of a classical simple Lie algebra $\mathfrak{g}$, Kostant's weight multiplicity formula gives the multiplicity of the weight $\mu$ in the irreducible representation with highest weight $\lambda$, which we denote by $m(\lambda,\mu)$. Kostant's weight multiplicity formula is an alternating sum over the Weyl group of the Lie algebra whose terms are determined via a vector partition function. The Weyl alternation set $\mathcal{A}(\lambda,\mu)$ is the set of elements of the Weyl group that contribute nontrivially to the multiplicity $m(\lambda,\mu)$. In this article, we prove that Weyl alternation sets are order ideals in the weak Bruhat order of the corresponding Weyl group. Specializing to the Lie algebra $\mathfrak{sl}_{r+1}(\mathbb{C})$, we give a complete characterization of the Weyl alternation sets $\mathcal{A}(\tilde{\alpha},\mu)$, where $\tilde{\alpha}$ is the highest root and $\mu$ is a negative root, answering a question of Harry posed in 2024. We also provide some enumerative results that pave the way for our future work, where we aim to prove Harry's conjecture that the $q$-analog of Kostant's weight multiplicity formula is $m_q(\tilde{\alpha},\mu)=q^{r+j-i+1}+q^{r+j-i}-q^{j-i+1}$ when $\mu=-(\alpha_i+\alpha_{i+1}+\cdots+\alpha_{j})$ is a negative root of $\mathfrak{sl}_{r+1}(\mathbb{C})$.

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arXiv2024

On a combinatorial puzzle arising from the theory of Lascoux polynomials

Lascoux polynomials are a class of nonhomogeneous polynomials which form a basis of the full polynomial ring. Recently, Pan and Yu showed that Lascoux polynomials can be defined as generating polynomials for certain collections of diagrams consisting of unit cells arranged in the first quadrant generated from an associated ``key diagram" by applying sequences of ``$K$-Kohnert moves". Within diagrams generated in this manner, certain cells are designated as special and referred to as ``ghost cells". Given a fixed Lascoux polynomial, Pan and Yu established a combinatorial algorithm in terms of ``snow diagrams" for computing the maximum number of ghost cells occurring in a diagram defining a monomial of the given polynomial; having this value allows for one to determine the total degree of the given Lascoux polynomial. In this paper, we study the combinatorial puzzle which arises when one replaces key diagrams by arbitrary diagrams in the definition of Lascoux polynomials. Specifically, given an arbitrary diagram, we consider the question of determining the maximum number of ghost cells contained within a diagram among those formed from our given initial one by applying sequences of $K$-Kohnert moves. In this regard, we establish means of computing the aforementioned max ghost cell value for various families of diagrams as well as for diagrams in general when one takes a greedy approach.

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arXiv2024

Contact Lie poset algebras of types B, C, and D

We extend a recently established combinatorial index formula applying to Lie poset algebras of types B, C, and D. Then, using the extended index formula, we determine a characterization of contact Lie poset algebras of types B, C, and D corresponding to posets of height one in terms of an associated graph.

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arXiv2023

On ranked and bounded Kohnert posets

In this paper, we explore combinatorial properties of the posets associated with Kohnert polynomials. In particular, we determine a sufficient condition guaranteeing when such ``Kohnert posets'' are bounded and two necessary conditions for when they are ranked. Moreover, we apply the aforementioned conditions to find complete characterizations of when Kohnert posets are bounded and when they are ranked in special cases, including those associated with Demazure characters.

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arXiv2023

Seaweed algebras and the unimodal spectrum property

If $\mathfrak{g}$ is a Frobenius Lie algebra, then the spectrum of $\mathfrak{g}$ is an algebraic invariant equal to the multiset of eigenvalues corresponding to a particular operator acting on $\mathfrak{g}$. In the case of Frobenius seaweed subalgebras of $A_{n-1}=\mathfrak{sl}(n)$, or type-A seaweeds for short, it has been shown that the spectrum can be computed combinatorially using an attendant graph. With the aid of such graphs, it was further shown that the spectrum of a type-A seaweed consists of an unbroken sequence of integers centered at $\frac{1}{2}$. It has been conjectured that if the eigenvalues are arranged in increasing order, then the sequence of multiplicities forms a unimodal sequence about $\frac{1}{2}$. Here, we establish this conjecture for certain families of Frobenius type-A seaweeds by finding explicit formulas for their spectra; in fact, for some families we are able to show that the corresponding sequences of multiplicities form log-concave sequences. All arguments are combinatorial.

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arXiv2022

Seaweed algebras

The index of a Lie algebra is an important algebraic invariant, but it is notoriously difficult to compute. However, for the suggestively-named seaweed algebras, the computation of the index can be reduced to a combinatorial formula based on the connected components of a "meander": a planar graph associated with the algebra. Our index analysis on seaweed algebras requires only basic linear and abstract algebra. Indeed, the main goal of this survey-type article is to introduce a broader audience to seaweed algebras with minimal appeal to specialized language and notation from Lie theory. This said, we present several results that do not appear elsewhere and do appeal to more advanced language in the Introduction to provide added context.

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arXiv2021

The breadth of Lie poset algebras

The breadth of a Lie algebra $L$ is defined to be the maximal dimension of the image of $ad_x=[x,-]:L\to L$, for $x\in L$. Here, we initiate an investigation into the breadth of three families of Lie algebras defined by posets and provide combinatorial breadth formulas for members of each family.

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arXiv2020

Contact Lie poset algebras

We provide a combinatorial recipe for constructing all posets of height at most two for which the corresponding type-A Lie poset algebra is contact. In the case that such posets are connected, a discrete Morse theory argument establishes that the posets' simplicial realizations are contractible. It follows from a cohomological result of Coll and Gerstenhaber on Lie semi-direct products that the corresponding contact Lie algebras are absolutely rigid.

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arXiv2020

Contact Seaweeds

A ($2k+1$)$-$dimensional contact Lie algebra is one which admits a one-form $\varphi$ such that $\varphi \wedge (d\varphi)^k\ne0$. Such algebras have index one, but this is not generally a sufficient condition. Here we show that index-one type-A seaweed algebras are necessarily contact. Examples, together with a method for their explicit construction, are provided.

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