arXiv subjects

Murray R. Bremner

Publications and source records attributed to Murray R. Bremner.

At least 19 recordsLinked to original sources

arXiv2026

Enumerating Multi-Operator Monomials in Commutative and Noncommutative Settings

We study enumeration problems for multi-operator monomials generated from one indeterminate by an associative multiplication together with finitely many unary operators. We consider four regimes, according to whether multiplication is commutative and whether the unary operators commute. In the case where the unary operators do not commute, we obtain explicit multigraded generating functions and coefficient formulas, including a multinomial refinement of the Narayana numbers, together with interpretations in terms of rooted trees, restricted lattice paths, and binary trees. When the unary operators commute, we derive canonical representatives and effective recurrences, with corresponding monotonicity conditions in the combinatorial models. When multiplication is commutative, the sequence decomposition is replaced by a multiset decomposition, leading to exp--log generating functions and Euler-transform recurrences. In special cases, the resulting sequences recover classical families including the Catalan numbers, the small Schröder numbers, and rooted-tree numbers.

math.CO↗

arXiv2026

The Veronese square of the dendriform operad

Veronese powers of operads were introduced in 2020 By Dotsenko, Markl, and Remm \cite{DMR}. The $m$-th Veronese power of a weight-graded operad $\mathcal{V}$ is the suboperad $\mathcal{V}^{[m]}$ generated by the operations of weight $m$. If $\mathcal{V}$ is generated by binary operations and governs the variety $\mathbf{V}$ of algebras, this gives a natural definition of the concept of $(m{+}1)$-ary $\mathbf{V}$-algebras. In particular, the Veronese square ($m=2$) corresponds to ternary algebras. We choose five generating operations for the Veronese square of the dendriform operad. We represent the dendriform operad as a suboperad of the Rota-Baxter operad, and express the quadratic relations satisfied by the generating operations as the kernel of a rewriting map. We use combinatorics of monomials and computational linear algebra to determine the kernel. We obtain 33 linearly independent quadratic relations satisfied by the Veronese square.

math.RA↗

arXiv2025

A new generalization of the Narayana numbers inspired by linear operators on associative $d$-ary algebras

We introduce and study a generalization of the Narayana numbers $N_d(n,k) = \frac{1}{n+1} \binom{n+1}{k+1} \binom{ n + (n-k)(d-2)+1}{k}$ for integers $d \geq 2$ and $n,k \geq 0$. This two-parameter array extends the classical Narayana numbers ($d=2$) and yields a $d$-ary analogue of the Catalan numbers $C_d(n) = \sum_{k=0}^n N_d(n,k)$. We give nine combinatorial interpretations of $N_d(n,k)$ that unify and generalize known combinatorial interpretations of the Narayana numbers and $C_3(n)$ in the literature. In particular, we show that $N_d(n,k)$ counts a natural class of operator monomials over a $d$-ary associative algebra, thereby extending a result of Bremner and Elgendy for the binary case. We also construct explicit bijections between these monomials and several families of classic combinatorial objects, including Schröder paths, Dyck paths, rooted ordered trees, and $231$-avoiding permutations.

math.CO↗

arXiv2025

Algebraic identities for linear operators on associative triple systems (long version)

We present the first classification of algebraic identities in 3 variables for linear operators on associative structures. We work in the context of associative triple systems, but since any associative algebra with product $xy$ becomes an associative triple system with product $xyz$, our results apply to associative algebras as well. This is the first time that Rota's classification problem for linear operators has been extended to algebras with an $n$-ary operation for $n \ge 3$. Our work is an application of computational linear algebra to the classification problem for linear operators. We begin with a generic operator identity with indeterminate coefficients. From this we use operadic partial compositions to derive a large sparse matrix whose nonzero entries are the indeterminates. We follow the rank principle which states that significant operator identities correspond to coefficients which produce submaximal rank of the matrix. For operator identities of multiplicity 1 (each term contains the operator once) we obtain 6 families with 1 parameter, and 1 isolated solution. For multiplicity 2, we obtain 6 families with 2 parameters, 27 families with 1 parameter, and 9 isolated solutions.

math.RA↗

arXiv2025

Enumeration and Asymptotic Formulas for Rectangular Partitions of the Hypercube

We study a two-parameter generalization of the Catalan numbers: $C_{d,p}(n)$ is the number of ways to subdivide the $d$-dimensional hypercube into $n$ rectangular blocks using orthogonal partitions of fixed arity $p$. Bremner \& Dotsenko introduced $C_{d,p}(n)$ in their work on Boardman--Vogt tensor products of operads; they used homological algebra to prove a recursive formula and a functional equation. We express $C_{d,p}(n)$ as simple finite sums, and determine their growth rate and asymptotic behaviour. We give an elementary proof of the functional equation, using a bijection between hypercube decompositions and a family of full $p$-ary trees. Our results generalize the well-known correspondence between Catalan numbers and full binary trees.

math.CO↗

arXiv2025

Operator identities of multiplicity 3 for associative algebras

We consider algebraic identities for linear operators on associative algebras in which each term has degree 2 (the number of variables) and multiplicity 3 (the number of occurrences of the operator). We apply the methods of earlier work by the author and Elgendy which classified operator identities of degree 2, multiplicities 1 and 2. We begin with the general operator identity of multiplicity 3 which has 10 terms and indeterminate coefficients. We use the operadic concept of partial composition to generate all consequences of this identity in degree 3, multiplicity 4. The coefficient matrix of these consequences has size $105 \times 20$ and indeterminate entries. We compute the partial Smith form of this matrix and use Gröbner bases for determinantal ideals to discover which values of the indeterminates produce a matrix of submaximal rank. The only possible submaximal values of the rank are 16 and 19: there are 6 new identities of rank 16, and 8 new identities of rank 19.

math.RA↗

arXiv2022

Higher polynomial identities for mutations of associative algebras

We study polynomial identities satisfied by the mutation product $xpy - yqx$ on the underlying vector space of an associative algebra $A$, where $p, q$ are fixed elements of $A$. We simplify known results for identities in degree $4$, proving that only two identities are necessary and sufficient to generate them all; in degree 5, we show that adding one new identity suffices; in degree 6, we demonstrate the existence of a number of new identities.

math.RA↗

arXiv2021

A new classification of algebraic identities for linear operators on associative algebras

We introduce a new approach to the classification of operator identities, based on basic concepts from the theory of algebraic operads together with computational commutative algebra applied to determinantal ideals of matrices over polynomial rings. We consider operator identities of degree 2 (the number of variables in each term) and multiplicity 1 or 2 (the number of operators in each term), but our methods apply more generally. Given an operator identity with indeterminate coefficients, we use partial compositions to construct a matrix of consequences, and then use computer algebra to determine the values of the indeterminates for which this matrix has submaximal rank. For multiplicity 1 we obtain six identities, including the derivation identity. For multiplicity 2 we obtain eighteen identities and two parametrized families, including the left and right averaging identities, the Rota-Baxter identity, the Nijenhuis identity, and some new identities which deserve further study.

math.RA↗

arXiv2018

Special Identities for Comtrans Algebras

Comtrans algebras, arising in web geometry, have two trilinear operations, commutator and translator. We determine a Gröbner basis for the comtrans operad, and state a conjecture on its dimension formula. We study multilinear polynomial identities for the special commutator $[x,y,z] = xyz-yxz$ and special translator $\langle x, y, z \rangle = xyz-yzx$ in associative triple systems. In degree 3, the defining identities for comtrans algebras generate all identities. In degree 5, we simplify known identities for each operation and determine new identities relating the operations. In degree 7, we use representation theory of the symmetric group to show that each operation satisfies identities which do not follow from those of lower degree but there are no new identities relating the operations. We use noncommutative Gröbner bases to construct the universal associative envelope for the special comtrans algebra of $2 \times 2$ matrices.

math.RA↗

arXiv2017

The Magic Star of Exceptional Periodicity

We present a periodic infinite chain of finite generalisations of the exceptional structures, including e8, the exceptional Jordan algebra (and pair), and the octonions. We demonstrate that the exceptional Jordan algebra is part of an infinite family of finite-dimensional matrix algebras (corresponding to a particular class of cubic Vinberg's T-algebras). Correspondingly, we prove that e8 is part of an infinite family of algebras (dubbed "Magic Star" algebras) that resemble lattice vertex algebras.

hep-th↗

arXiv2016

One-parameter deformations of the diassociative and dendriform operads

Livernet and Loday constructed a polarization of the nonsymmetric associative operad A with one operation into a symmetric operad SA with two operations (the Lie bracket and Jordan product), and defined a one-parameter deformation of SA which includes Poisson algebras. We combine this with the dendriform splitting of an associative operation into the sum of two nonassociative operations, and use Koszul duality for quadratic operads, to construct one-parameter deformations of the nonsymmetric dendriform and diassociative operads into the category of symmetric operads.

math.QA↗

arXiv2015

Quadratic nonsymmetric quaternary operads

We use computational linear algebra and commutative algebra to study spaces of relations satisfied by quadrilinear operations. The relations are analogues of associativity in the sense that they are quadratic (every term involves two operations) and nonsymmetric (every term involves the identity permutation of the arguments). We focus on determining those quadratic relations whose cubic consequences have minimal or maximal rank. We approach these problems from the point of view of the theory of algebraic operads.

math.RA↗

arXiv2013

Structure of the rational monoid algebra for Boolean matrices of order 3

We use computer algebra to study the 512-dimensional associative algebra Q B_3, the rational monoid algebra of 3 x 3 Boolean matrices. We obtain a basis for the radical in bijection with the 42 non-regular elements of B_3. The center of the 470-dimensional semisimple quotient has dimension 14; we use a splitting algorithm to find a basis of orthogonal primitive idempotents. We show that the semisimple quotient is the direct sum of simple two-sided ideals isomorphic to d x d rational matrix algebras for d = 1, 1, 1, 2, 3, 3, 3, 3, 6, 6, 7, 9, 9, 12. We construct the irreducible representations of B_3 over Q by calculating the representation matrices for a minimal set of generators.

math.RA↗

arXiv2013

Dendriform analogues of Lie and Jordan triple systems

We use computer algebra to determine all the multilinear polynomial identities of degree $\le 7$ satisfied by the trilinear operations $(a \cdot b) \cdot c$ and $a \cdot (b \cdot c)$ in the free dendriform dialgebra, where $a \cdot b$ is the pre-Lie or the pre-Jordan product. For the pre-Lie triple products, we obtain one identity in degree 3, and three independent identities in degree 5, and we show that every identity in degree 7 follows from the identities of lower degree. For the pre-Jordan triple products, there are no identities in degree 3, five independent identities in degree 5, and ten independent irreducible identities in degree 7. Our methods involve linear algebra on large matrices over finite fields, and the representation theory of the symmetric group.

math.RA↗

arXiv2013

A polynomial identity for the bilinear operation in Lie-Yamaguti algebras

We use computer algebra to demonstrate the existence of a multilinear polynomial identity of degree 8 satisfied by the bilinear operation in every Lie-Yamaguti algebra. This identity is a consequence of the defining identities for Lie-Yamaguti algebras, but is not a consequence of anticommutativity. We give an explicit form of this identity as an alternating sum over all permutations of the variables in a polynomial with 8 terms. Our computations also show that such identities do not exist in degrees less than 8.

math.RA↗

arXiv2013

Free associative algebras, noncommutative Grobner bases, and universal associative envelopes for nonassociative structures

These are the lecture notes from my short course of the same title at the CIMPA Research School on Associative and Nonassociative Algebras and Dialgebras: Theory and Algorithms - In Honour of Jean-Louis Loday (1946-2012), held at CIMAT, Guanajuato, Mexico, February 17 to March 2, 2013. The underlying motivation is to apply the theory of noncommutative Grobner bases in free associative algebras to the construction of universal associative envelopes for nonassociative structures defined by multilinear operations. Trilinear operations were classified by the author and Peresi in 2007. In her Ph.D. thesis of 2012, Elgendy studied the universal associative envelopes of nonassociative triple systems obtained by applying these trilinear operations to the 2-dimensional simple associative triple system. In these notes I use computer algebra to extend some aspects of her work to the 4-dimensional and 6-dimensional simple associative triple systems.

math.RA↗

arXiv2012

Special identities for the pre-Jordan product in the free dendriform algebra

Pre-Jordan algebras were introduced recently in analogy with pre-Lie algebras. A pre-Jordan algebra is a vector space $A$ with a bilinear multiplication $x \cdot y$ such that the product $x \circ y = x \cdot y + y \cdot x$ endows $A$ with the structure of a Jordan algebra, and the left multiplications $L_\cdot(x)\colon y \mapsto x \cdot y$ define a representation of this Jordan algebra on $A$. Equivalently, $x \cdot y$ satisfies these multilinear identities: [see PDF]. The pre-Jordan product $x \cdot y = x \succ y + y \prec x$ in any dendriform algebra also satisfies these identities. We use computational linear algebra based on the representation theory of the symmetric group to show that every identity of degree $\le 7$ for this product is implied by the identities of degree 4, but that there exist new identities of degree 8 which do not follow from those of lower degree. There is an isomorphism of $S_8$-modules between these new identities and the special identities for the Jordan diproduct in an associative dialgebra.

math.RA↗

arXiv2012

On Kruskal's theorem that every 3 x 3 x 3 array has rank at most 5

In the first part of this paper, we consider 3 x 3 x 3 arrays with complex entries, and provide a complete self-contained proof of Kruskal's theorem that the maximum rank is 5. In the second part, we provide a complete classification of the canonical forms of 3 x 3 x 3 arrays over F_2; in particular, we obtain explicit examples of such arrays with rank 6.

math.RA↗