Joseph P. S. Kung

Invariant theory, Young bitableaux, and combinatorics

Publisher Summary Since its emergence in the middle of the last century, the invariant theory has oscillated between two clearly distinguishable poles. The first, and the one that was later to survive the temporary death of the field, is geometry. Invariants were identified with the invariants of surfaces. Their study, the aim was which to give information about the solution of systems of polynomial equations, was to lead to the rise of commutative algebra. The second pole of the invariant theory was algorithmic. All invariant theory is ultimately concerned with the problem to generalize to tensors the eigenvalue theory of matrices. This chapter provides a self-contained combinatorial presentation of the vector invariant theory over an arbitrary infinite field. This is done by proving the Straightening Formula, which is one of the fundamental algorithms of multilinear algebra. The straightening formula has two advantages. First, it holds over the ring of integers. Second, it recognizes the crucial role played by the notion of a bitableau in obtaining a characteristic-free proof of the first fundamental theorem.

The cycle structure of a linear transformation over a finite field

Abstract In analogy with the cycle decomposition of a permutation, we study the enumerative properties of the decomposition of an invertible linear transformation into a direct sum of cyclic linear transformations. Our main tool is a vector space analog of the Polya cycle index.

Bimatroids and invariants

A bimatroid B between the sets S and T incorporates the combinatorial exchange properties of relative invariants of the special linear group acting on a vector space and its dual, or equivalently, when S and T are finite, the exchange properties fo nonsingular minors of a matrix whose columns and rows are indexed by S and T. A rather simple idea, basically that of adjoining an identity matrix, yields a construction which produces, from the bimatroid B, a matroid R on the disjoint union S ∪ T which encodes all the structure of B. This gives a method of translating matroidal concepts and results into the language of bimatroids. We also define an analog of matrix multiplication for bimatroids: This operation generalizes matroid induction and affords another combinatorial interpretation of a linear transformation.

The radon transforms of a combinatorial geometry, I

Abstract Over a field of characteristic zero, the rank of the point-copoint incidence matrix of a combinatorial geometry of rank ⩾ 2 equals the number of points. The proof uses a finite analogue of the Radon transform.

Growth rates of minor-closed classes of matroids

For a minor-closed class M of matroids, let h(k) denote the maximum number of elements in a simple rank-k matroid in M. We prove that, if M does not contain all simple rank-2 matroids, then h(k) is finite and is either linear, quadratic, or exponential.

On the Rédei zeta function

Abstract Let L be a locally finite lattice. An order function ν on L is a function defined on pairs of elements x, y (with x ≤ y) in L such that ν(x, y) = ν(x, z) ν(z, y). The Redei zeta function of L is given by ϱ(s; L) = Σx∈Lμ(O, x) ν(O, x)−s. It generalizes the following functions: the chromatic polynomial of a graph, the characteristic polynomial of a lattice, the inverse of the Dedekind zeta function of a number field, the inverse of the Weil zeta function for a variety over a finite field, Philip Halls φ-function for a group and Redeis zeta function for an abelian group. Moreover, the paradigmatic problem in all these areas can be stated in terms of the location of the zeroes of the Redei zeta function.

Matchings and radon transforms in lattices. I. Consistent lattices

An element in a lattice is join-irreducible if x=a∨b implies x=a or x=b. A meet-irreducible is a join-irreducible in the order dual. A lattice is consistent if for every element x and every join-irreducible j, the element x∨j is a join-irreducible in the upper interval [x, î]. We prove that in a finite consistent lattice, the incidence matrix of meet-irreducibles versus join-irreducibles has rank the number of join-irreducibles. Since modular lattices and their order duals are consistent, this settles a conjecture of Rival on matchings in modular lattices.

A classification of modularly complemented geometric lattices

A geometric lattice G is said to be modularly complemented if for every point in G , there exists a modular copoint not containing it. We prove that a connected modularly complemented geometric lattice of rank at least four is either a Dowling lattice or the lattice of flats of a projective geometry with some of its points deleted.

Convolution-multiplication identities for Tutte polynomials of graphs and matroids

We give a general convolution-multiplication identity for the multivariate and bivariate rank generating polynomial of a graph or matroid. The bivariate rank generating polynomial is transformable to and from the Tutte polynomial by simple algebraic operations. Several identities, almost all already known in some form, are specializations of this identity. Combinatorial or probabilistic interpretations are given for the specialized identities.

Combinatorial geometries representable over GF(3) and GF(q). II. Dowling geometries

Letq be an odd prime power not divisible by 3. In Part I of this series, it was shown that the number of points in a rank-n combinatorial geometry (or simple matroid) representable over GF(3) and GF(q) is at mostn2. In this paper, we show that, with the exception ofn = 3, a rank-n geometry that is representable over GF(3) and GF(q) and contains exactlyn2 points is isomorphic to the rank-n Dowling geometry based on the multiplicative group of GF(3).