arXiv subjects

David J. Hemmer

Publications and source records attributed to David J. Hemmer.

At least 19 recordsLinked to original sources

arXiv2026

Billiard Orbits in Young Diagrams: Medial Links, Bicycle Spaces, and Domino Tilings

We study diagonal billiard trajectories inside the Young diagram of an integer partition $λ$. A trajectory has slope $\pm 1$, passes straight through sides shared by adjacent cells, and reflects from the exterior boundary until it closes. Let $σ(λ)$ be the number of closed orbits. This extends the mirror-curve model of Chokwe sona sand drawings studied by Gerdes from rectangular grids to arbitrary Young diagrams. Let $G_λ$ be the cell-adjacency graph of $λ$, with its natural planar embedding. We identify the billiard orbits with the components of the medial link of $G_λ$, and deduce that $σ(λ) = 1 + \dim \mathcal{B}(G_λ) = \mathrm{nullity}\, L(G_λ)$ over $\mathbb{F}_2$, where $\mathcal{B}$ is the binary bicycle space and $L$ the mod-2 Laplacian. Writing $λ^\square$ for the diagram obtained by deleting the first row and column of $λ$, we further prove $σ(λ) = 1 + \mathrm{nullity}_{\mathbb{F}_2} A(G_{λ^\square})$. For rectangles this recovers Gerdes' formula $σ(n^m) = \gcd(m,n)$ via identities for Fibonacci polynomials over $\mathbb{F}_2$, and in general it gives the characterization: $σ(λ) = 1$ if and only if $λ^\square$ has an odd number of domino tilings. Using the checkerboard bipartition of $λ^\square$, we decompose $σ(λ) - 1$ into a color-imbalance term, related to the BG-rank of Berkovich-Garvan, and an even rank-deficiency term. This yields parity restrictions and lower bounds for the orbit number, and shows that for any fixed $d$, asymptotically all partitions have more than $d$ orbits. We also prove that $σ(λ)$ is at most the Durfee length of $λ$, determine $σ(n, n-1, \ldots, 1) = \lceil n/2 \rceil$ for staircase partitions, and show that the adjacency-nullity formula is independent of the ground field.

math.CO↗

arXiv2026

Equal knapsack identities between symmetric group character degrees

We prove a series of ``knapsack'' type equalities for irreducible character degrees of symmetric groups. That is, we find disjoint subsets of the partitions of $n$ so that the two corresponding character-degree sums are equal. Our main result refines our recent description of the Riordan numbers as the sum of all character degrees $f^λ$ where $λ$ is a partition of $n$ into three parts of the same parity. In particular, the sum of the ``fat-hook'' degrees $f^{(k,k,1^{n-2k})}+f^{(k+1,k+1,1^{n-2k-2})}$ equals the sum of all $f^λ$ where $λ$ has three parts, with the second equal to $k$ and the second and third of equal parity. We further prove an infinite family of additional ``knapsack'' identities between character degrees

math.CO↗

arXiv2026

Specht Modules With Trivial Direct Summands

We prove that, in characteristic $2$, a Specht module indexed by a nonhook partition cannot have a trivial direct summand. The proof uses the KLR grading and a recent graded homomorphism theorem of Hudak. Together with Murphy's theorem for hooks, this gives the complete classification of trivial direct summands of Specht modules in characteristic $2$, answering a question posed by Collins--Dodge and Dodge--Van Vlack.

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arXiv2026

New columns in decomposition matrices of symmetric groups for every block

The central unsolved problem in the modular representation theory of symmetric groups is to find the decomposition matrices, which describe how irreducible representations in characteristic zero decompose upon reduction modulo a prime characteristic $p$. In this paper we determine a large number of new columns in these decomposition matrices, namely those labeled by partitions whose $p$-divisible hooks have all even arm lengths. In particular in odd characteristic $p$, for every possible block of every possible symmetric group $S_n$, we determine at least one complete column. These columns are multiplicity-free and are described by a recently introduced combinatorial statistic of partitions (depending on $p$), called the odd sequence. As an application, we determine the indecomposable summands of Foulkes modules $H^{(2^m)}$.

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arXiv2026

Divisible Arm Lengths, Crystal Reflections, and Enumeration of Newly Found Decomposition Columns

In recent work the authors determine complete columns of symmetric-group decomposition matrices in odd prime characteristic $p$ labeled by $p$-regular partitions for which every hook of length divisible by $p$ has even arm length. In the present paper we enumerate these partitions and prove that each block of $p$-weight $w$ contains precisely \[ \binom{w+\frac{p-3}{2}}{w} \] such partitions. More generally, for any integers $d,e>1$, we study and enumerate $d$-balanced $e$-regular partitions -- partitions for which every hook of length divisible by $e$ has arm length divisible by $d$. Our first main result is that the crystal (affine) reflections preserve the $d$-balanced property for all $d,e > 1$. It follows that, for fixed $d$, $e$, and $w$, the number of $d$-balanced $e$-regular partitions in a block of $e$-weight $w$ is independent of the $e$-core. We then compute this number by working in RoCK blocks, obtaining an explicit binomial formula valid for every block. We also investigate closely related odd sequences of partitions. Among others, we find the generating function of the number of odd sequences occurring in a block. Alongside their representation-theoretic relevance, we expect these results to be of independent combinatorial interest.

math.CO↗

arXiv2026

Optimal Play, Nontransitivity, and Nash Equilibria in Dice Bingo

We study Dice Bingo, a game in which players fill a $3\times3$ bingo board whose entries are possible sums of two fair dice. After each roll, a player marks one matching square, and the goal is to complete a row, column, or diagonal. We model optimal play for a fixed board as a finite Markov decision process and derive Bellman equations that compute the exact expected number of rolls required to obtain a bingo. Using this framework, we identify a unique optimal board up to natural symmetries and determine its exact expected completion time. We then investigate head-to-head competition in which two players observe the same sequence of dice rolls. By analyzing a joint Markov chain that tracks both boards simultaneously, we compute (in exact arithmetic) win, loss, and tie probabilities. Surprisingly, a board with a worse expected completion time can nevertheless be favored in head-to-head competition. Motivated by this phenomenon, we exhibit nontransitive triples of bingo boards: board $A$ is favored against board $B$, board $B$ is favored against board $C$, and board $C$ is favored against board $A$. Finally, we consider strategic play in which players adapt their choices to their opponent's board rather than merely minimizing their own completion time. In this setting, optimal decisions depend on the opponent's state, leading naturally to game-theoretic analysis. We present a position with no pure Nash equilibrium and compute an explicit mixed Nash equilibrium.

math.HO↗

arXiv2025

Partition-theoretic model of prime distribution

We make an application of ideas from partition theory to a problem in multiplicative number theory. We propose a deterministic model of prime number distribution, from first principles related to properties of integer partitions, that naturally predicts the prime number theorem as well as the twin prime conjecture. The model posits that, for $n\geq 2$, $$p_{n}\ =\ 1\ +\ 2\sum_{j=1}^{n-1}\left\lceil \frac{d(j)}{2}\right\rceil\ +\ \varepsilon(n),$$ where $p_k$ is the $k$th prime number, $d(k)$ is the divisor function, and $\varepsilon(k)$ is an explicit error term that is negligible asymptotically; both the main term and error term represent enumerative functions in our conceptual model. We refine the error term to give numerical estimates of $π(n)$ similar to those provided by the logarithmic integral, and much more accurate than $\operatorname{li}(n)$ up to $n=10{,}000$ where the estimates are {\it almost exact}. We then perform computational tests of unusual predictions of the model, finding limited evidence of predictable variations in prime gaps.

math.NT↗

arXiv2025

New Identities in the Character Table of Symmetric Groups involving Riordan Numbers

Amdeberhan recently proposed certain equalities between sums in the character table of symmetric groups. These equalities are between signed column sums in the character table, summing over the rows labeled by partitions in $\Ev$, where $λ$ is a partition of $n$ with $r$ nonzero parts and $\Ev$ is a multiset containing $2^r$ partitions of $2n$. While we observe that these equalities are not true in general, we prove that they do hold in interesting special cases. These lead to new equalities between sums of degrees of irreducible characters for the symmetric group and a new combinatorial interpretation for the Riordan numbers in terms of degrees of irreducible characters labeled by partitions with three parts of the same parity. This is the first, to our knowledge, theorem about degrees of symmetric group characters with parity conditions imposed on the partitions indexing the characters.

math.CO↗

arXiv2025

Generating functions for fixed points of the Mullineux map

Mullineux defined an involution on the set of $e$-regular partitions of $n$. When $e=p$ is prime, these partitions label irreducible symmetric group modules in characteristic $p$. Mullineux's conjecture, since proven, was that this ``Mullineux map" described the effect on the labels of taking the tensor product with the one-dimensional signature representation. Counting irreducible modules fixed by this tensor product is related to counting irreducible modules for the alternating group $A_n$ in prime characteristic. In 1991, Andrews and Olsson worked out the generating function counting fixed points of Mullineux's map when $e=p$ is an odd prime (providing evidence in support of Mullineux's conjecture). In 1998, Bessenrodt and Olsson counted the fixed points in a $p$-block of weight $w$. We extend both results to arbitrary $e$, and determine the corresponding generating functions. When $e$ is odd but not prime the extension is immediate, while $e$ even requires additional work and the results, which are different, have not appeared in the literature.

math.CO↗

arXiv2024

Palindrome Partitions and the Calkin-Wilf Tree

There is a well-known bijection between finite binary sequences and integer partitions. Sequences of length r correspond to partitions of perimeter r+1. Motivated by work on rational numbers in the Calkin-Wilf tree, we classify partitions whose corresponding binary sequence is a palindrome. We give a generating function that counts these partitions, and describe how to efficiently generate all of them. Atypically for partitions generating functions, we find an unusual significance to prime degrees. Specifically, we prove there are nontrivial palindrome partitions of n except when n=3 or n+1 is prime. We find an interesting new "branching diagram" for partitions, similar to Young's lattice, with an action of the Klein four group corresponding to natural operations on the binary sequences.

math.CO↗

arXiv2024

Partitions with fixed points in the sequence of first-column hook lengths

Recently, Blecher and Knopfmacher applied the notion of fixed points to integer partitions. This has already been generalized and refined in various ways such as $h$-fixed points for an integer parameter $h$ by Hopkins and Sellers. Here, we consider the sequence of first column hook lengths in the Young diagram of a partition and corresponding fixed hooks. We enumerate these, using both generating function and combinatorial proofs, and find that they match occurrences of part sizes equal to their multiplicity. We establish connections to work of Andrews and Merca on truncations of the pentagonal number theorem and classes of partitions partially characterized by certain minimal excluded parts (mex).

math.CO↗

arXiv2015

The Lie module and its complexity

The complexity of a module is an important homological invariant that measures the polynomial rate of growth of its minimal projective resolution. For the symmetric group $Σ_n$, the Lie module $\mathsf{Lie}(n)$ has attracted a great deal of interest in recent years. We prove here that the complexity of $\mathsf{Lie}(n)$ in characteristic $p$ is $t$ where $p^t$ is the largest power of $p$ dividing $n$, thus proving a conjecture of Erdmann, Lim and Tan. The proof uses work of Arone and Kankaanrinta which describes the homology $\operatorname{H}_\bullet(Σ_n, \mathsf{Lie}(n))$ and earlier work of Hemmer and Nakano on complexity for modules over $Σ_n$ that involves restriction to Young subgroups.

math.GR↗

arXiv2012

"Frobenius twists" in the representation theory of the symmetric group

For the general linear group $GL_n(k)$ over an algebraically closed field $k$ of characteristic $p$, there are two types of "twisting" operations that arise naturally on partitions. These are of the form $λ\rightarrow pλ$ and $λ\rightarrow λ+ p^rτ$ The first comes from the Frobenius twist, and the second arises in various tensor product situations, often from tensoring with the Steinberg module. This paper surveys and adds to an intriguing series of seemingly unrelated symmetric group results where this partition combinatorics arises, but with no structural explanation for it. This includes cohomology of simple, Specht and Young modules, support varieties for Specht modules, homomorphisms between Specht modules, the Mullineux map, $p$-Kostka numbers and tensor products of Young modules.

math.RT↗

arXiv2009

On the cohomology of Young modules for the symmetric group

The main result of this paper is an application of the topology of the space $Q(X)$ to obtain results for the cohomology of the symmetric group on $d$ letters, $Σ_d$, with `twisted' coefficients in various choices of Young modules and to show that these computations reduce to certain natural questions in representation theory. The authors extend classical methods for analyzing the homology of certain spaces $Q(X)$ with mod-$p$ coefficients to describe the homology $\HH_{\bullet}(Σ_d, V^{\otimes d})$ as a module for the general linear group $GL(V)$ over an algebraically closed field $k$ of characteristic $p$. As a direct application, these results provide a method of reducing the computation of $\text{Ext}^{\bullet}_{Σ_{d}}(Y^λ,Y^μ)$ (where $Y^λ$, $Y^μ$ are Young modules) to a representation theoretic problem involving the determination of tensor products and decomposition numbers. In particular, in characteristic two, for many $d$, a complete determination of $\Hs Y^λ)$ can be found. This is the first nontrivial class of symmetric group modules where a complete description of the cohomology in all degrees can be given. For arbitrary $d$ the authors determine $\HH^i(Σ_d,Y^λ)$ for $i=0,1,2$. An interesting phenomenon is uncovered--namely a stability result reminiscent of generic cohomology for algebraic groups. For each $i$ the cohomology $\HH^i(Σ_{p^ad}, Y^{p^aλ})$ stabilizes as $a$ increases. The methods in this paper are also powerful enough to determine, for any $p$ and $λ$, precisely when $\HH^{\bullet}(\sd,Y^λ)=0$. Such modules with vanishing cohomology are of great interest in representation theory because their support varieties constitute the representation theoretic nucleus.

math.RT↗

arXiv2009

A combinatorial approach to Specht module cohomology

For a Specht module S^λfor the symmetric group Σ_d, the cohomology H^i(Σ_d, S^λ) is known only in degree i=0. We give a combinatorial criterion equivalent to the nonvanishing of the degree i=1 cohomology, valid in odd characteristic. Our condition generalizes James' solution in degree zero. We apply this combinatorial description to give some computations of Specht module cohomology, together with an explicit description of the corresponding modules. Finally we suggest some general conjectures that might be particularly amenable to proof using this description.

math.RT↗

arXiv2009

Cohomology and generic cohomology of Specht modules for the symmetric group

Cohomology of Specht modules for the symmetric group can be equated in low degrees with corresponding cohomology for the Borel subgroup B of the general linear group GL_d(k), but this has never been exploited to prove new symmetric group results. Using work of Doty on the submodule structure of symmetric powers of the natural GL_d(k) module together with work of Andersen on cohomology for B and its Frobenius kernels, we prove new results about H^i(Σ_d, S^λ). We recover work of James in the case i=0. Then we prove two stability theorems, one of which is a "generic cohomology" result for Specht modules equating cohomology of S^{pλ} with S^{p^2λ}. This is the first theorem we know relating Specht modules S^λand S^{pλ}. The second result equates cohomology of S^λwith S^{λ+ p^aμ} for large a.

math.RT↗

arXiv2008

The complexity of certain Specht modules for the symmetric group

During the 2004-2005 academic year the VIGRE algebra research group at the University of Georgia computed the complexities of certain Specht modules S^λfor the symmetric group, using the computer algebra program Magma. The complexity of an indecomposable module does not exceed the p-rank of the defect group of its block. The Georgia group conjectured that, generically, the complexity of a Specht module attains this maximal value; that it is smaller precisely when the Young diagram of $λ$ is built out of $p \times p$ blocks. We prove one direction of this conjecture by showing these Specht modules do indeed have less than maximal complexity. It turns out that this class of partitions, which has not previously appeared in the literature, arises naturally as the solution to a question about the $p$-weight of partitions and branching.

math.RT↗

arXiv2006

Symmetric group modules with Specht and dual Specht filtrations

The author and Nakano recently proved that multiplicities in a Specht filtration of a symmetric group module are well-defined precisely when the characteristic is at least five. This result suggested the possibility of a symmetric group theory analogous to that of good filtrations and tilting modules for $GL_n(k)$. This paper is an initial attempt at such a theory. We obtain two sufficient conditions that ensure a module has a Specht filtration, and a formula for the filtration multiplicities. We then study the categories of modules that satisfy the conditions, in the process obtaining a new result on Specht module cohomology. Next we consider symmetric group modules that have both Specht and dual Specht filtrations. Unlike tilting modules for $GL_n(k)$, these modules need not be self-dual, and there is no nice tensor product theorem. We prove a correspondence between indecomposable self-dual modules with Specht filtrations and a collection of $GL_n(k)$-modules which behave like tilting modules under the tilting functor. We give some evidence that indecomposable self-dual symmetric group modules with Specht filtrations may be self-dual trivial source modules.

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