arXiv subjects

Darij Grinberg

Publications and source records attributed to Darij Grinberg.

At least 19 recordsLinked to original sources

arXiv2026

Powers of matrices with all principal minors equal to 1

We say that a square matrix $A$ is \emph{$1$-principled} if all its principal minors are equal to $1$. We show that over any well-behaved ring, any power $A^m$ of a $1$-principled matrix $A$ is again $1$-principled. Well-behaved rings include all reduced rings as well as all quotients of commutative rings modulo integrally closed ideals; in particular, all fields and all quotients of $\mathbb{Z}$ are well-behaved. We note that $m$ can be any integer, positive or negative. This generalizes Problem B5 of the 2021 Putnam contest in multiple directions. Over arbitrary commutative rings, we identify a stronger property that is always inherited by powers: We say that a matrix $A = \left(a_{i,j}\right)_{i,j\in\left[n\right]}$ is \emph{$1$-nullcyclic} if all its diagonal entries are $1$ and if all the cyclic products $a_{i_1, i_2} a_{i_2, i_3} \cdots a_{i_k, i_1}$ with $k>1$ and distinct $i_1,i_2,\ldots,i_k$ vanish. We show that if $A$ is $1$-nullcyclic, then so is $A^m$ for any integer $m$. Furthermore, every $1$-nullcyclic matrix is $1$-principled over any commutative ring, while the converse holds if the ring is well-behaved. Along the way, we prove analogous results that don't require the diagonal entries to be $1$. These are concerned with \emph{principled matrices} (those whose principal minors equal the respective products of diagonal entries) and \emph{nullcyclic matrices} (those whose cyclic products $a_{i_1, i_2} a_{i_2, i_3} \cdots a_{i_k, i_1}$ with $k>1$ and distinct $i_1,i_2,\ldots,i_k$ vanish); their diagonal entries can be arbitrary. A crucial auxiliary result, which holds for any $n\times n$-matrix $A$, is that the cyclic products $a_{i_1, i_2} a_{i_2, i_3} \cdots a_{i_k, i_1}$ (with $k>1$ and distinct $i_1,i_2,\ldots,i_k$) are integral over the ideal generated by the principal minors of $A$ minus the corresponding products of diagonal entries of $A$.

math.AC↗

arXiv2026

Detropicalization as a proof technique

Rational functions make sense over any commutative ring, as long as the denominators are invertible. When there is no subtraction involved, they even apply over semirings (rings without subtraction). It is particularly worthwhile to evaluate them over the *tropical semiring*, in which the roles of addition and multiplication are played by maxima and addition. Over this semiring, algebraic results often acquire combinatorial meaning. We give a few examples.

math.CO↗

arXiv2026

The V/L recursion for Macdonald's 7th Variation Schur polynomials

We generalize and prove the recursive relation \[ S_{\lambda}(V) = \sum_{L\subseteq V\text{ line}} S_{\lambda}(V \mathbin{/\mkern-5mu/} L) \] conjectured by I. G. Macdonald for his "7th variation" of the Schur functions. This variation is a family of polynomials over a finite field that mimic the (straight and skew) Schur polynomials using powers of the Frobenius.

math.CO↗

arXiv2026

Compositions of $n$-homomorphisms

We study $n$-homomorphisms in the sense of Khudaverdian--Voronov, but generalized to maps from arbitrary rings to arbitrary commutative rings. We show that the sum of an $n$-homomorphism and an $m$-homomorphism is an $\left( n+m\right) $-homomorphism, and that the composition of an $n$-homomorphism and an $m$-homomorphism is an $nm$-homomorphism. The proofs are entirely combinatorial.

math.RA↗

arXiv2026

A constructive proof of Orzech's theorem

Let $A$ be a commutative ring with unity, and $M$ a finitely generated $A$-module. In 1971, Morris Orzech showed that any surjective $A$-module homomorphism from a submodule of $M$ to $M$ must be an isomorphism. We give a constructive proof of this fact using the Cayley--Hamilton theorem.

math.AC↗

arXiv2026

Noncommutative Abel-like identities

We generalize the Abel--Hurwitz identities to an almost entirely noncommutative setting. Namely, let $V$ be a finite set of size $n$, and let $\mathbb{L}$ be any noncommutative ring. For each $s\in V$, let $x_{s}\in\mathbb{L}$. Set $x\left( S\right) :=\sum_{s\in S}x_{s}$ for any $S\subseteq V$. Let $X$ and $Y$ be two elements of $\mathbb{L}$ such that $X+Y$ lies in the center of $\mathbb{L}$. Then, we show that% \begin{align*} & \sum_{S\subseteq V}\left( X+x\left( S\right) \right) ^{\left\vert S\right\vert }\left( Y-x\left( S\right) \right) ^{n-\left\vert S\right\vert }=\sum_{\substack{i_{1},i_{2},\ldots,i_{k}\in V\text{ distinct}% }}\left( X+Y\right) ^{n-k}x_{i_{1}}x_{i_{2}}\cdots x_{i_{k}};\\ & \sum_{S\subseteq V}X\left( X+x\left( S\right) \right) ^{\left\vert S\right\vert -1}\left( Y-x\left( S\right) \right) ^{n-\left\vert S\right\vert }=\left( X+Y\right) ^{n};\\ & \sum_{S\subseteq V}X\left( X+x\left( S\right) \right) ^{\left\vert S\right\vert -1}\left( Y-x\left( S\right) \right) ^{n-\left\vert S\right\vert -1}\left( Y-x\left( V\right) \right) =\left( X+Y-x\left( V\right) \right) \left( X+Y\right) ^{n-1}. \end{align*} (Negative powers are understood to be cancelled by other factors.)

math.CO↗

arXiv2026

The left-to-right minima basis of the group algebra of the symmetric group (updated version)

We introduce a new basis of the group algebra of the symmetric group, built using the left-to-right minima sets of permutations. We show that on this basis, the descent algebra acts by triangular operators, thus making it an analogue of a cellular basis. The proof involves Dynkin elements (nested commutators) of the free algebra and their interactions with the $\mathbf B$-basis.

math.CO↗

arXiv2025

The trace Cayley-Hamilton theorem

In this expository paper, various properties of matrix traces, determinants and adjugate matrices are proved, including the *trace Cayley-Hamilton theorem*, which says that \[ kc_k + \sum_{i=1}^k \operatorname{Tr} (A^i) c_{k-i} = 0 \qquad \text{for every } k\in\mathbb{N} \] whenever $A$ is an $n\times n$-matrix with characteristic polynomial $\det (tI_n - A) = \sum_{i=0}^n c_{n-i} t^i$ over a commutative ring $\mathbb{K}$. While the results are not new, some of the proofs are. The proofs illustrate some general techniques in linear algebra over commutative rings.

math.RA↗

arXiv2025

Top to random and reverse: analysis of a new descent algebra shuffle

We study the "top-to-random-and-reverse shuffle", defined as the top-to-random shuffle in the symmetric group algebra composed with the permutation $w_0$ (which sends each $i$ to $n+1-i$). More generally, we analyze the composition of any B-basis element of the descent algebra with $w_0$. We show that the minimal polynomial of any such composition (over $\mathbb{Q}$) factors into distinct linear factors, which correspond to the "signed knapsack numbers" of set compositions. This is a counterpart to an analogous property of the B-basis elements themselves, which was proved by Brown using Bidigare's face monoid. In the case of the top-to-random-and-reverse shuffle, the minimal polynomial turns out to be $\prod_{k \in \set{-n+2} \cup \interval{-n+4, n-3} \cup \set{0} \cup \set{n}} \tup{x-k}$.

math.CO↗

arXiv2025

The representation theory of somewhere-to-below shuffles

The *somewhere-to-below shuffles* are the elements \[ t_{\ell} := \operatorname{cyc}_{\ell}+\operatorname{cyc}_{\ell,\ell+1}+\operatorname{cyc}_{\ell,\ell+1,\ell+2}+\cdots+\operatorname{cyc}_{\ell,\ell+1,\ldots,n} \] (for $\ell \in \{1,2,\dots,n\}$) in the group algebra $\mathbf{k}[S_n]$ of the $n$-th symmetric group $S_n$. Their linear combinations are called the *one-sided cycle shuffles*. We determine the eigenvalues of the action of any one-sided cycle shuffle on any Specht module $\mathcal{S}^{\lambda}$ of $S_n$.

math.CO↗

arXiv2025

An introduction to the algebra of rings and fields

This is an introduction to rings and fields, written for a quarter-long undergraduate course. It includes the basic properties of ideals, modules, algebras and polynomials, the constructions of ring extensions and finite fields, some number-theoretical applications (such as a proof of quadratic reciprocity and Jacobsthal's formulas for $p = x^2 + y^2$), and tastes of Gr\"obner bases and the Smith normal form. Familiarity with groups and vector spaces is assumed, though no deep results from either theory are used. Over 200 exercises are included (mostly without solutions).

math.RA↗

arXiv2025

Rook sums in the symmetric group algebra

Let $\mathcal{A}$ be the group algebra $\mathbf{k}[S_n]$ of the $n$-th symmetric group $S_n$ over a commutative ring $\mathbf{k}$. For any two subsets $A$ and $B$ of $[n]$, we define the elements \[ \nabla_{B,A}:=\sum_{\substack{w\in S_n;\\w\left( A\right) =B}} w \qquad \text{and} \qquad \widetilde{\nabla}_{B,A}:=\sum_{\substack{w\in S_n;\\w\left( A\right) \subseteq B}}w \] of $\mathcal{A}$. We study these elements, showing in particular that their minimal polynomials factor into linear factors (with integer coefficients). We express the product $\nabla_{D,C}\nabla_{B,A}$ as a $\mathbb{Z}$-linear combination of $\nabla_{U,V}$'s. More generally, for any two set compositions (i.e., ordered set partitions) $\mathbf{A}$ and $\mathbf{B}$ of $\left\{ 1,2,\ldots,n\right\} $, we define $\nabla_{\mathbf{B},\mathbf{A}}\in\mathcal{A}$ to be the sum of all permutations $w\in S_n$ that send each block of $\mathbf{A}$ to the corresponding block of $\mathbf{B}$. This generalizes $\nabla_{B,A}$. The factorization property of minimal polynomials does not extend to the $\nabla_{\mathbf{B},\mathbf{A}}$, but we describe the ideal spanned by the $\nabla_{\mathbf{B},\mathbf{A}}$ and a further ideal complementary to it. These two ideals have a "mutually annihilative" relationship, are free as $\mathbf{k}$-modules, and appear as annihilators of tensor product $S_n$-representations; they are also closely related to Murphy's cellular bases, Specht modules, pattern-avoiding permutations and even some algebras appearing in quantum information theory.

math.CO↗

arXiv2025

An equality for balanced digraphs

Consider a directed multigraph $D$ that is balanced (i.e., at each vertex, the indegree equals the outdegree). Let $A$ be its set of arcs. Fix an integer $k$. Let $s$ be a vertex of $D$. We show that the number of $k$-element subsets $B$ of $A$ that contain no cycles but contain a path from each vertex to $s$ (we call them "$s$-convergences") is independent on $s$. This generalizes known facts about spanning arborescences, acyclic orientations and maximal acyclic subdigraphs (or, equivalently, minimum feedback arc sets). Moreover, this result can be generalized even further, replacing "contain no cycles" with "have a given set of cycles".

math.CO↗

arXiv2025

An introduction to the symmetric group algebra

This is an introduction to the group algebras of the symmetric groups, written for a quarter-long graduate course. After recalling the definition of group algebras (and monoid algebras) in general, as well as basic properties of permutations, we introduce several families of elements in the symmetric group algebras $\mathbf{k}[S_n]$ such as the Young--Jucys--Murphy elements, the (sign-)integrals and the conjugacy class sums. Then comes a chapter on group actions and representations in general, followed by the core of this text: a study of the representations of symmetric groups (i.e., of left $\mathbf{k}[S_n]$-modules), including the classical theory of Young tableaux and Young symmetrizers. We prove in detail the main facts including the characterization of irreducible representations (in characteristic $0$), the Garnir relations, the standard basis theorem, the description of duals of Specht modules, and the hook length formula, as well as a number of less known results. Finally, we describe several bases of $\mathbf{k}[S_n]$ that arise from the study of Specht modules, including the Murphy cellular bases. The methods used are elementary and computational. We aim to assume as little as possible of the base ring $\mathbf{k}$, and to use as little as possible from representation theory (nothing more advanced than Maschke and Jordan--H\"older). Over 100 exercises (without solutions) are scattered through the text.

math.CO↗

arXiv2025

An Introduction to Algebraic Combinatorics

This is an introduction to algebraic combinatorics, written for a quarter-long graduate course. It starts with a rigorous introduction to formal power series with some combinatorial applications, then discusses integer partitions (proving Jacobi's triple product identity), permutations (Lehmer codes, cycles) and subtractive methods (alternating sums, cancellations and inclusion-exclusion principles, with a particular focus on sign-reversing involutions and determinants). The last chapter introduces symmetric polynomials and proves the Littlewood--Richardson rule using Bender--Knuth involutions (a la Stembridge). The appendix contains over 200 exercises (without solutions).

math.CO↗

arXiv2025

A permutation based approach to the $q$-deformation of the Dynkin Operator

Introduced by Solomon, the descent algebra is a significant subalgebra of the group algebra of the symmetric group $\mathbf{k}S_n$ related to many important algebraic and combinatorial topics. It contains all the classical Lie idempotents of $\mathbf{k}S_n$, in particular the Dynkin operator, a fundamental tool for studying the free Lie algebra. We look at a $q$-deformation of the Dynkin operator and study its action over the descent algebra with classical combinatorial tools like Solomon's Mackey formula. This leads to elementary proofs that the operator is indeed an idempotent for $q=1$ as well as to interesting formulas and algebraic structures especially when $q$ is a root of unity.

math.CO↗

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.

math.CO↗

arXiv2024

The Dowker theorem via discrete Morse theory

The Dowker theorem is a classical result in the topology of finite spaces, claiming that any binary relation between two finite spaces defines two homotopy-equivalent complexes (the Dowker complexes). Recently, Barmak strengthened this to a simple-homotopy-equivalence. We reprove Barmak's result using a combinatorial argument that constructs an explicit acyclic matching in the sense of discrete Morse theory.

math.CO↗