Orders of torsion units of integral reality-based algebras with rational multiplicities

A reality-based algebra (RBA) is a finite-dimensional associative algebra with involution over [Formula: see text] whose distinguished basis [Formula: see text] contains [Formula: see text] and is closed under pseudo-inverse. An integral RBA is one whose structure constants in its distinguished basis are integers. If the algebra has a one-dimensional representation taking positive values on [Formula: see text], then we say that the RBA has a positive degree map. These RBAs have a standard feasible trace, and the multiplicities of the irreducible characters in the standard feasible trace are the multiplicities of the RBA. In this paper, we show that for integral RBAs with positive degree map whose multiplicities are rational, any finite subgroup of torsion units whose elements are all of degree [Formula: see text] and have algebraic integer coefficients must have order dividing a certain positive integer determined by the degree map and the multiplicities. The paper concludes with a thorough investigation of the properties of RBAs that force multiplicities to be rational.

G-GRAPHS AND SPECIAL REPRESENTATIONS FOR BINARY DIHEDRAL GROUPS IN GL(2,ℂ)

Given a finite subgroup G⊂GL(2,ℂ), it is known that the minimal resolution of singularity ℂ2/G is the moduli space Y=G-Hilb(ℂ2) of G-clusters ⊂ℂ2. The explicit description of Y can be obtained by calculating every possible distinguished basis for as vector spaces. These basis are the so-called G-graphs. In this paper we classify G-graphs for any small binary dihedral subgroup G in GL(2,ℂ), and in the context of the special McKay correspondence we use this classification to give a combinatorial description of special representations of G appearing in Y in terms of its maximal normal cyclic subgroup H ⊴ G.

Comultiplication Rules for the Double Schur Functions and Cauchy Identities

The double Schur functions form a distinguished basis of the ring $\Lambda(x\!\parallel\!a)$ which is a multiparameter generalization of the ring of symmetric functions $\Lambda(x)$. The canonical comultiplication on $\Lambda(x)$ is extended to $\Lambda(x\!\parallel\!a)$ in a natural way so that the double power sums symmetric functions are primitive elements. We calculate the dual Littlewood–Richardson coefficients in two different ways thus providing comultiplication rules for the double Schur functions. We also prove multiparameter analogues of the Cauchy identity. A new family of Schur type functions plays the role of a dual object in the identities. We describe some properties of these dual Schur functions including a combinatorial presentation and an expansion formula in terms of the ordinary Schur functions. The dual Littlewood–Richardson coefficients provide a multiplication rule for the dual Schur functions.