Yasuo Teranishi

The ring of invariants of matrices

This action of G defines an action of G on an algebra C[W] = C[xo(l), • , Xij(l)] of all polynomial functions on W. We denote by C[W] the subalgebra of G invariant polynomials. This is a finitely generated subalgebra of C[W]. If Z = 1 it is a classical result that this ring of invariants is a polynomial ring in n variables. In fact the coefficients of characteristic polynomial of the matrix X(ΐ) — (x^(l)) are algebraically independent invariants and the ring of invariants is generated by them. By the Newtons formula all coefficients of characteristic polynomial of X(ί) are expressed by n traces Tr(X(l)), Ίx(X\ΐj), . . . ,Tr(X(l)»),

Noncommutative Invariant Theory

Throughout this article K will be a field of characteristic zero. Let V be a finite dimensional vector space over K. Let

The Hoffman number of a graph

T(V) = K \oplus V \oplus V^{ \otimes 2} \oplus \ldots

On the Ring of Simultaneous Invariants for the Gleason-MacWilliams Group

denote the tensor algebra over V. The group of K-automorphisms GL(V) can be identified with the group of homogeneous automorphisms of T(V). If G is a subgroup of GL(V), there is an induced homogeneous action of G on T(V).

Equitable switching and spectra of graphs

An eigenvalue of a graph is said to be a main eigenvalue if it has an eigenvector not orthogonal to the main vector j=(1,1,…,1). In this paper we shall study some properties of main eigenvalues of a graph.

Laplacian spectra and invariants of graphs

For a connected graph G with n vertices, let {λ1,λ2,.....,λr} be the set of distinct positive eigenvalues of the Laplacian matrix of G. The Hoffman number µ(G) of G is defined by µ(G)= λ1 λ2 ... λr/n. In this paper, we study some properties and applications of the Hoffman number.

Eigenvalues and automorphisms of a graph

In this paper we study the number of spanning forests of a graph. Let G be a connected simple graph. (1) We give a lower bound for the number of spanning forests of G in terms of the edge connectivity of G. (2) We give an upper bound for the number of rooted spanning forests of G. (3) We describe the elementary symmetric functions of inverse positive Laplacian eigenvalues of a tree. (4) We determine all Laplacian integral graphs with prime number of spanning trees. (5) We give a simple proof of a theorem of K. Hashimoto on Ihara zeta function.

Main eigenvalues and automorphisms of a graph

We construct a canonical generating set for the polynomial invariants of the simultaneous diagonal action (of arbitrary number of l factors) of the two-dimensional finite unitary reflection group G of order 192, which is called the group No. 9 in the list of Shephard and Todd, and is also called the Gleason?MacWilliams group. We find this canonical set in the vector space (?i=1lV)G, where V denotes the (dual of the) two-dimensional vector space on which the group G acts, by applying the techniques of Weyl (i.e., the polarization process of invariant theory) to the invariants Cx, y ]G0of the two-dimensional group G0of order 48 which is the intersection of G and SL(2, C). It is shown that each element in this canonical set corresponds to an irreducible representation which appears in the decomposition of the action of the symmetric group Sl. That is, by letting the symmetric group Slacts on each element of the canonical generating set, we get an irreducible subspace on which the symmetric group Slacts irreducibly, and all these irreducible subspaces give the decomposition of the whole space (?i=1lV)G. This also makes it possible to find the generating set of the simultaneous diagonal action (of arbitrary l factors) of the group G. This canonical generating set is different from the homogeneous system of parameters of the simultaneous diagonal action of the group G. We can construct Jacobi forms (in the sense of Eichler and Zagier) in various ways from the invariants of the simultaneous diagonal action of the group G, and our canonical generating set is very fit and convenient for the purpose of the construction of Jacobi forms.