MacMahon Master Theorem Research Articles

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.

Immanants, Schur Functions, and the Macmahon Master Theorem

The relationship between the immanant and the Schur symmetric function is examined. Two expressions for the immanant are given in terms of the determinant. Generalisations include Foata and Zeilberger’s $\beta$-extension of the MacMahon Master theorem. The relationships to some little known results of Littlewood and to idempotents constructed by Young are given.

Immanants, Schur functions, and the MacMahon master theorem

The relationship between the immanant and the Schur symmetric function is examined. Two expressions for the immanant are given in terms of the determinant. Generalisations include Foata and Zeilberger’s $\beta$-extension of the MacMahon Master theorem. The relationships to some little known results of Littlewood and to idempotents constructed by Young are given.

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.

Heaps of Pieces, I: Basic Definitions and Combinatorial Lemmas

We introduce the combinatorial notion of heaps of pieces, which gives a geometric interpretation of the Cartier-Foata's commutation monoid. This theory unifies and simplifies many other works in Combinatorics : bijective proofs in matrix algebra (MacMahon Master theorem, inversion matrix formula, Jacobi identity, Cayley-Hamilton theorem), combinatorial theory for general (formal) orthogonal polynomials, reciprocal of Rogers-Ramanujan identities, graph theory (matching and chromatic polynomials). Heaps may bring new light on classical subjects as poset theory. They are related to other fields as Theoretical Computer Science (parallelism) and Statistical Physics (directed animals problem, lattice gas model with hard-core interactions). Complete proofs and definitions are given in sections 2, 3,4,5. Other sections give a summary of possible applications of heaps.

Applications of the theory of automata in enumeration

A general method for applying techniques from the theory of automata to enumeration problems is developed, and several applications are given. These include the enumeration of reduced words in a finite group action and a new proof of the MacMahon Master Theorem.

Expansion formulas. I. A general method

Using the MacMahon Master Theorem, Carlitz, in ( SIAM J. Appl. Math. 26 (1974), 431–436; 8 (1977), 320–336) obtains the expansion ∑ n i ⩾ 0 ( n ̄ i + b 1) n1 … ( n ̄ s + b s) n s (u 1 n i … u s n s (n 1! … n s!) = e b 1x 1 + ⋯ b sx s det(δ ij − x ia ij), where for j = 1, 2,…, s, -n j = ∑ i = 1 s n i a ij and for i = 1, 2,…, s, u i = x i exp(−∑ j = 1 s a ijx j) . A “factorial” analog of this expansion and other related results are also given. In this paper we derive one general formula from which these expanions follow as special cases. This formula demonstrates a bijection between a class of expansion formulas and Sheffer sequences of polynomials.

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.