MacMahon Master Research Articles

Algebraic Combinatorics on Trace Monoids: Extending Number Theory to Walks on Graphs

Partially commutative monoids provide a powerful tool to study graphs, viewingwalks as words whose letters, the edges of the graph, obey a specific commutation rule. A particularclass of traces emerges from this framework, the hikes, whose alphabet is the set of simple cycleson the graph. We show that hikes characterize undirected graphs uniquely, up to isomorphism, andsatisfy remarkable algebraic properties such as the existence and uniqueness of a prime factorization.Because of this, the set of hikes partially ordered by divisibility hosts a plethora of relations in directcorrespondence with those found in number theory. Some applications of these results are presented,including a permanantal extension to MacMahon's master theorem and a derivation of the Ihara zetafunction.

A note on element centralizers in finite Coxeter groups

Abstract The normalizer NW (WJ ) of a standard parabolic subgroup WJ of a finite Coxeter group W splits over the parabolic subgroup with complement NJ consisting of certain minimal length coset representatives of WJ in W. In this note we show that (with the exception of a small number of cases arising from a situation in Coxeter groups of type Dn ) the centralizer CW (w) of an element w ∈ W is in a similar way a semidirect product of the centralizer of w in a suitable small parabolic subgroup WJ with complement isomorphic to the normalizer complement NJ . Then we use this result to give a new short proof of Solomon's Character Formula and discuss its connection to MacMahon's master theorem.

Zeons, Permanents, the Johnson Scheme, and Generalized Derangements

Starting with the zero-square “zeon algebra,” the connection with permanents is shown. Permanents of submatrices of a linear combination of the identity matrix and all-ones matrix lead to moment polynomials with respect to the exponential distribution. A permanent trace formula analogous to MacMahon's master theorem is presented and applied. Connections with permutation groups acting on sets and the Johnson association scheme arise. The families of numbers appearing as matrix entries turn out to be related to interesting variations on derangements. These generalized derangements are considered in detail as an illustration of the theory.

Generating Functions for Hecke Algebra Characters

Abstract Certain polynomials in n2 variables that serve as generating functions for symmetric group characters are sometimes called (Sn) character immanants. We point out a close connection between the identities of Littlewood–Merris–Watkins and Goulden–Jackson, which relate Sn character immanants to the determinant, the permanent and MacMahon's Master Theorem. From these results we obtain a generalization of Muir's identity. Working with the quantum polynomial ring and the Hecke algebra Hn(q), we define quantum immanants that are generating functions for Hecke algebra characters. We then prove quantum analogs of the Littlewood–Merris–Watkins identities and selected Goulden–Jackson identities that relate Hn(q) character immanants to the quantum determinant, quantum permanent, and quantum Master Theorem of Garoufalidis–Lê–Zeilberger. We also obtain a generalization of Zhang's quantization of Muir's identity.

A Quantization of a theorem of Goulden and Jackson

A theorem of Goulden and Jackson which gives interesting formulae for character immanants also implies MacMahon's Master Theorem. We quantize Goulden and Jackson's theorem to give formulae for quantum character immanants in such a way as to obtain a known quantization of MacMahon's Master Theorem due to Garoufalidis-Lê-Zeilberger. In doing so, we also quantize formulae of Littlewood, Merris and Watkins concerning induced character immanants. Le Théorème Master de MacMahon peut être vu comme un corollaire d'un résultat de Goulden et Jackson décrivant certaines formules satisfaites par les caractères immanants. Dans cet article, on obtient une version quantique des formules de Goulden et Jackson, ce qui permet de donner une nouvelle preuve du résultat de Garoufalidis, Lê et Zeilberger sur la quantification du Théorème Master de MacMahon. Ce faisant, on arrive aussi à quantifier les formules de Littlewood, Merris et Watkins concernant les caractères immanants induits.

Koszul algebras and the quantum MacMahon master theorem

We give a new proof of the quantum version of MacMahon's master theorem due to Garoufalidis, Lê and Zeilberger (in the one-parameter case) and to Konvalinka and Pak (in the multiparameter case) by deriving it from known facts about Koszul algebras.

A new proof of the Garoufalidis–Lê–Zeilberger quantum MacMahon Master Theorem

We propose a new proof of the quantum version of MacMahon's Master Theorem, established by Garoufalidis, Lê and Zeilberger.

Extension of MacMahon's Master Theorem to pre-semi-rings

An extended version of the classical MacMahon's “Master Theorem” to algebraicstructures much more general than the field of real numbers, namely semi-rings and even pre-semi-rings, is proposed. Due to the fact that many semi-rings or pre-semi-rings are not additive groups, such concepts as the determinant of a matrix cannot be used any more, and the statement of the result itself has to be significantly different. When specialized to the field of real numbers, however, the extended result is shown to reduce to the classical version of MacMahon's “Master Theorem”.

MacMahon's Master Theorem, Representation Theory, and Moments of Wishart Distributions

D. Foata and D. Zeilberger (1988, SIAM J. Discrete Math.4, 425–433) and D. Vere-Jones (1988, Linear Algebra Appl.111, 119–124) independently derived a generalization of MacMahon's master theorem. In this article we apply their result to obtain an explicit formula for the moments of arbitrary polynomials in the entries of X, a real random matrix having a Wishart distribution. In the case of the complex Wishart distributions, the same method is applicable. Furthermore, we apply the representation theory of GL(d,C), the complex general linear group, to derive explicit formulas for the expectation of Kronecker products of any complex Wishart random matrix.

Angular momentum theory, umbral calculus, and combinatorics

The theory of the unitary irreducible representations of the unitary group SU(2) is reviewed with the objective of demonstrating the fundamental role that the umbral calculus and combinatorics play. For the physicist, the study of the group SU(2) is synonymous with the theory of angular momentum in quantum theory, and Kronecker products of irreducible unitary representations comprise the mathematical apparatus for building composite systems from simpler constituents. The Clebsch-Gordon coefficients, which are essential to the binary theory of composite systems, are shown to have an umbral calculus origin. MacMahon's master theorem is shown to be the basic result for generating the mathematical quantities needed for bringing comprehension to the properties of composite systems.

A new multidimensional matrix inverse with applications to multiple q-series

We compute the inverse of a specific infinite r-dimensional matrix, extending a matrix inverse of Krattenthaler. Our inversion is different from the r-dimensional matrix inversion recently found by Schlosser but generalizes a multidimensional matrix inversion previously found by Chu. As applications of our matrix inversion we derive some multidimensional q-series identities. Among these are q-analogues of Carlitz' multidimensional Abel-type expansion formulas. Furthermore, we derive a q-analogue of MacMahon's Master Theorem.

The Combinatorics of a Class of Representation Functions

In the study of the irreducible representations of the unitary groupU(n), one encounters a class of polynomials defined onn2indeterminateszij, 1⩽i,j⩽n, which may be arranged into ann×nmatrix arrayZ=(zij). These polynomials are indexed by double Gelfand patterns, or equivalently, by pairs of column strict Young tableaux of the same shape. Using the double labeling property, one may define a square matrixD(Z), whose elements are the double-indexed polynomials. These matrices possess the remarkable “group multiplication property”D(XY)=D(X)D(Y) for arbitrary matricesXandY, even though these matrices may be singular. ForZ=U∈U(n), these matrices give irreducible unitary representations ofU(n). These results are known, but not always fully proved from the extensive physics literature on representation of the unitary groups, where they are often formulated in terms of the boson calculus, and the multiplication property is unrecognized. The generality of the multiplication property is the key to understanding group representation theory from the purview of combinatorics. The combinatorial structure of the general polynomials is expected to be intricate, and in this paper, we take the first step to explore the combinatorial aspects of a special class which can be defined in terms of the set of integral matrices with given row and column sums. These special polynomials are denoted byLα,β(Z), whereαandβare integral vectors representing the row sums and column sums of a class of integral matrices. We present a combinatorial interpretation of the multiplicative properties of these polynomials. We also point out the connections with MacMahon's Master Theorem and Schwinger's inner product formula, which is essentially equivalent to MacMahon's Master Theorem. Finally, we give a formula for the double Pfaffian, which is crucial in the studies of the generating function of the 3n−jcoefficients in angular momentum theory. We also review the background of the general polynomials and give some of their properties.

Determinant, permanent, and MacMahon's master theorem

As a unified extension of determinant and permanent, a new matrix function with two extra parameters is introduced. Basic formulas are established, which contain several known relations between determinants and permanents as special cases. Some expansions in terms of the matrix functions are investigated, which yield MacMahon's master theorem, its dual version, and their common generalization.

MacMahon's Master Theorem, Double Tableau Polynomials, and Representations of Groups

It is shown that MacMahon's master theorem gives the diagonal elements of a class of irreducible representations of the general linear group,Gl(n,C), and hence the trace of these representations, or the group characters. These representations are unitary under restriction to the unitary subgroup and constitute the so-called totally symmetric representations. A generalization of MacMahon's master theorem that generates the elements of the representation matrix themselves is given. These relations for groups are, in fact, more general in that they apply to arbitrary matrices of indeterminates. These results are proved by using the properties of a class of Young–Weyl tableau polynomials defined overn2indeterminates. Forn=2, two sets of basic polynomials are defined in terms of Young–Weyl tableaux. The first set corresponds to the unitary irreducible representations of U(2), as defined in the physics literature and whose properties comprise a substantive part of what is known as the quantum theory of angular momentum; the second set is Rota's double standard tableau polynomials. These two sets of polynomials span the same space, and a comprehensive set of relations between them is given. A further generalization of MacMahon's master theorem pertaining to the general polynomial representations ofGL(n,C), as enumerated by partitions, is conjectured and is shown to contain the classical Schur function generating function as a special case.

On some classes of inverse series relations and their applications

This paper gives a brief exposition of several recent results obtained by the authors concerning some classes of inverse relations, including the general binomial-type inversions and two kinds of general Stirling reciprocal transforms. Also described are some applications of the related inversion techniques to combinatorics (including new proofs of Rogers-Ramanujan identities and MacMahon's master theorem), interpolation methods and certain problems related to special polynomials and number sequences.

The β-Extension of the Multivariable Lagrange Inversion Formula

We show that Gessel's combinatorial proof of the multivariable Lagrange inversion formula can be given a ,β-extension, which generalizes Foata and Zeilberger's, β-extension of MacMahon's master theorem. Moreover, we show that there is no need to use Jacobi's identity in the derivation of the Lagrange formula. Finally, combining Gessel's method and ours, we obtain a new proof of Jacobi's identity.

Some Multifold Reciprocal Transformations with Applications to Series Expansions

By means of a difference operation, a pair of reciprocal formulas is established which can be regarded as a kind of multifold analogue of Gould-Hsu inversions. Its rotated forms are used to derive some series expansions. One of them implies MacMahon's master theorem.

Weighted permutation problems and Laguerre polynomials

The connection between integrals of products of Laguerre polynomials, power series coefficients of certain rational functions of several variables, and certain numbers of weighted permutation problems is investigated. A combinatorial proof of our main result would be very desirable since this could lead the way to more general result, q-analogs, and perhaps even a q-analog of MacMahon's Master Theorem.