Explicit formulas for 2-characters

Ganter and Kapranov associated a 2-character to 2-representations of a finite group. Elgueta classified 2-representations in the category of 2-vector spaces 2Vectk in terms of cohomological data. We give an explicit formula for the 2-character in terms of this cohomological data and derive some consequences.     

On the geometry of the slice of trace-free {{SL_2(\mathbb{C})}}-characters of a knot group

Let K be a knot in an integral homology 3-sphere ?? with exterior E _K , and let B ₂ denote the two-fold branched cover of ?? branched along K. We construct a map ?? from the slice of trace-free ${{{\rm SL}_2(\mathbb{C})}}$ -characters of ?? ₁(E _K ) to the ${{{\rm SL}_2(\mathbb{C})}}$ -character variety of ?? ₁(B ₂). When this map is surjective, it describes the slice as the two-fold branched cover over the ${{{\rm SL}_2(\mathbb{C})}}$ -character variety of B ₂ with branched locus given by the abelian characters, whose preimage is precisely the set of metabelian characters. We show that each metabelian character can be represented as the character of a binary dihedral representation of ?? ₁(E _K ). The map ?? is shown to be surjective for all 2-bridge knots and all pretzel knots of type (p, q, r). An extension of this framework to n-fold branched covers is also described.     

Rings of SL2( $\Bbb{C}$ )-characters and the Kauffman bracket skein module

Let M be a compact orientable 3-manifold. The set of characters of SL ₂()-representations of forms a closed affine algebraic set. We show that its coordinate ring is isomorphic to a specialization of the Kauffman bracket skein module, modulo its nilradical. This is accomplished by realizing the module as a combinatorial analog of the ring in which tools of skein theory are exploited to illuminate relations among characters. We conclude with an application, proving that a small manifold's specialized module is necessarily finite dimensional. Received: April 18, 1996     

The combinatorics of a three-line circulant determinant

We study the polynomial , where ω is a primitivepth root of unity. This polynomial arises in CR geometry [1]. We show that it is the determinant of thep×p circulant matrix whose first row is (1, −x,0,…,0,−y,0,…,0), the −y being in positionq+1. Therefore, the coefficients of this polynomial Φ are integers that count certain classes of permutations. We show that all of the permutations that contribute to a fixed monomialx ^ry^s in Φ have the same sign, and we determine that sign. We prove that a monomialx ^ry^s appears in Φ if and only ifp dividesr+sq. Finally, we show that the size of the largest coefficient of the monomials in Φ grows exponentially withp, by proving that the permanent of the circulant whose first row is (1, 1, 0, …, 0, 1, 0, …, 0) is the sum of the absolute values of the monomials in the polynomial Φ. Supported by NSF Postdoctoral research grants.     

The automorphism group of a split metacyclic 2-group

This paper finds the order, structure and presentation for the automorphism group of any split metacyclic 2-group. Received: 2 August 2006     

The group PSL(2,q) is not the multiplication group of a loop

ABSTRACT

We introduce the notion of weak transitivity for torsion-free abelian groups. A torsion-free abelian group G is called weakly transitive if for any pair of elements x, y ∈ G and endomorphisms ?, ψ ∈ End(G) such that x? = y, yψ = x, there exists an automorphism of G mapping x onto y. It is shown that every suitable ring can be realized as the endomorphism ring of a weakly transitive torsion-free abelian group, and we characterize up to a number-theoretical property the separable weakly transitive torsion-free abelian groups.     

The 2-modular characters of the Fischer group

We determine the 2-modular character table of the Fischer group Fi₂₃.     

The determinant of a class of skew-symmetric toeplitz matrices

The determinant of the n×n matrix associated with the finite-difference operator μδ^2r+1 is obtained explicitly for all n and all integral r?0. An interesting combinatorial identity results.     

The permutation class group of a finite group

Analogously to the projective class group, the permutation class group of a finite group π can be defined as the group of equivalence classes of direct summands of integral permutation modules modulo permutation modules. It is shown that this group behaves nicely with respect to localization and completion, which then is used to prove that contrary to the projective class group - it is not always a torsion group. More precisely, the rank of the permutation class of group is computed.     

On the nilpotency class of a multiplicative group of a modular group algebra of a dihedral 2-group

It is proved that the wreath product of a second-order group and the commutant of a dihedral group is imbedded into a multiplicative group of a modular group algebra of a dihedral group of order 2 ⁿ . This implies that the nilpotency class of the multiplicative group is equal to 2 ^n–2, i.e., to the order of the commutant of the dihedral group.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 1, pp. 39–45, January, 1995.     

The weighted complexity and the determinant functions of graphs

The complexity of a graph can be obtained as a derivative of a variation of the zeta function [S. Northshield, A note on the zeta function of a graph, J. Combin. Theory Ser. B 74 (1998) 408-410] or a partial derivative of its generalized characteristic polynomial evaluated at a point [D. Kim, H.K. Kim, J. Lee, Generalized characteristic polynomials of graph bundles, Linear Algebra Appl. 429 (4) (2008) 688-697]. A similar result for the weighted complexity of weighted graphs was found using a determinant function [H. Mizuno, I. Sato, On the weighted complexity of a regular covering of a graph, J. Combin. Theory Ser. B 89 (2003) 17-26]. In this paper, we consider the determinant function of two variables and discover a condition that the weighted complexity of a weighted graph is a partial derivative of the determinant function evaluated at a point. Consequently, we simply obtain the previous results and disclose a new formula for the complexity from a variation of the Bartholdi zeta function. We also consider a new weighted complexity, for which the weights of spanning trees are taken as the sum of weights of edges in the tree, and find a similar formula for this new weighted complexity. As an application, we compute the weighted complexities of the product of the complete graphs.