Chih-Han Sah

Scissors congruences, II

In general, Y(X) is the scissors congruence group of polytopes in the space X. Unless stated explicitly, the group of motions of X is understood to be the group of all isometries of X. _@ is the extended hyperbolic n-space; it is obtained by adding to the hyperbolic n-space x”’ all the ideal points lying on 65~~. The geometry of a.F is that of conformal geometry on a sphere of dimension n 1. The group 4~~) captures the scissors congruence problem in a precise manner. On the other hand, the stable scissors congruence group p(_

Commuting transfer matrices for the four-state self-dual chiral Potts model with a genus-three uniformizing fermat curve

‘“) is more maneuverable, see Sah [19].

Cohomology of split group extensions, II

Abstract We consider the diagonal-to-diagonal transfer matrix of the four-state self-dual chiral Potts model on a square lattice. We show that there exists a family of models parametrized by a spectral variable u and an auxiliary variable b such that the commutation requirement [ T ( u , b ), T ( u ′, b )] = 0 needed for the existence of an infinite number of commuting constants of the motion holds. This family is uniformized by the curve x 4 + y 4 = 1 for all values of b except 0, 1 and ∞, where b = 1 corresponds to the critical self-dual model of Fateev and Zamolodchikov.

Group Cohomology and Hecke Operators

They introduced a spectral sequence relating the cohomology of G (with a suitable filtration) to the cohomologies of G/K and K. In later developments, Charlap and Vasquez [5, 61 showed that, under suitable restrictions, d, of the Hochschild-Serre spectral sequence can be described as the cup product with certain “characteristic” cohomology classes. In the special case where K is a free abelian group L of finite rank N and when G splits over K by r = G/K, these characteristic classes v 2t lie in H2(r, fP1(L, H,(L, Z))). Through straightforward, though formidable, procedures, Charlap and Vasquez found a number of interesting results concerning these characteristic classes. However, these characteristic classes still appeared mysterious and difficult to handle. The purpose of the present paper is to present a procedure to determine the most important one of these characteristic classes. To be precise, Charlap and Vasquez showed that vZt E H2(F, fPl(L, H,(L, Z))) have orders dividing 2; vsl = 0 and vpt = 0 holds for all t if and only if v22 = 0. They also showed that vZt are functorial in r so that vZt are universal when .F = GL(N, Z). We show that H2(GL(N, Z), HI(L, H,(L, Z))) is a group of order 2 for N > 2, and when N > 2, its generator is the universal characteristic class. Along the way, we calculate a number of cohomology groups similar to the one described. Our results and procedures can be applied to study the torsion of the total space of certain fibered varieties over symmetric space with abelian varieties for fibers (cf. Kuga [14]). Th ese applications will appear in a subsequent paper. Some of the cohomology group calculations are also useful in the study of finite simple groups. Indeed, one of the phenomena that persists throughout

Homology of classical Lie groups made discrete. II. H2, H3, and relations with scissors congruences☆

Consider the Taylor expansion of the infinite product:

A Generalized Leapfrog for Fullerene Structures

t_{n \ge 1}^\pi {(1 - {t^n})^2}{(1 - {t^{11n}})^2} = \sum\limits_{n \ge 1} {{a_n}{t^n}}

Hilbert's third problem : scissors congruence

.

Symmetric bilinear forms and quadratic forms

The present work extends a number of earlier results; see [S, 7, 15-181. For example, the following fundamental exact sequence (essentially due to Bloch and Wigner in somewhat different form, but not published by them) can be found in DuPont and Sah [7]: o~o/z-tH3(SL(2,@))~~(@)~n:(a=~)~~,(a=)~o. (2.12) The group Y(C) is defined in (1.4))( 1.8) and is isomorphic to the group PC considered in DuPont and Sah [7]. Also, the study of the scissors congruence problem had led to two exact sequences in DuPont [S, Theorems 1.3 and 1.43 that roughly correspond to the f -eigenspaces of (2.12) under the action of complex conjugation: O~A~H,(SU(2))~~S3/L~[WO([W/Z)-*H,(SU(2))-tO; (0.1)