A problem that arose in the study of the mass of the neutrino led us to the evaluation of a constant term with a variety of ramifications into several areas from Invariant Theory, Representation Theory, the Theory of Symmetric Functions and Combinatorics. A significant by-product of our evaluation is the construction of a trigraded Cohen Macaulay basis for the Invariants under an action of SL n...

Lascoux stated that the type A Kostka-Foulkes polynomials K?,?(t) expand positively in terms of so-called atomic polynomials. For any semisimple Lie algebra, former polynomial is a t-analogue multiplicity dominant weight ? irreducible representation highest ?. We formulate decomposition arbitrary type, and view it as strengthening monotonicity K?,?(t). also define combinatorial version decompos...

This work studies the remarkable relationships that hold among certain m-tuples of the Garsia-Haiman modules M and corresponding elements of the Macdonald basis. We recall that in 10], M is deened for a partition`n, as the linear span of derivatives of a certain bihomogeneouspolynomial (x; y) in the variables x 1 ; x 2 conjecturedin 6], 10] that M has n! dimensionsand that its bigraded Frobeniu...

The whole enterprise of spin compositions can be recast as simple enumerative combinatoric problems. We show here that enumerative combinatorics (EC)\citep{book:Stanley-2011} is a natural setting for spin composition, and easily leads to very general analytic formulae -- many of which hitherto not present in the literature. Based on it, we propose three general methods for computing spin multip...

The “D” matrices for all states of the two fundamental representations and octet are shown in the generalized Euler angle parameterization. The raising and lowering operators are given in terms of linear combinations of the left invariant vector fields of the group manifold in this parameterization. Using these differential operators the highest weight state of an arbitrary irreducible represen...

The Clebsch-Gordan formulas for d-function are studied for both nonunitary finite dimensional representation and unitary infinite dimensional representation of the SU (1, 1) group. It is shown that, in addition to the usual Clebsch-Gordan formulas for the above-mentioned two representations, there exists an additional type of Clebsch-Gordan fornmla connecting the nonunitary and unitary d-functi...

We study aspects of the propagation of strings on BTZ black holes. After performing a careful analysis of the global spacetime structure of generic BTZ black holes, and its relation to the geometry of the SL(2, R) group manifold, we focus on the simplest case of the massless BTZ black hole. We study the SL(2, R) Wess-Zumino-Witten model in the worldsheet minisuperspace limit, taking into accoun...

This paper characterizes the first cohomology group H(G,M) where M is a Banach space (with norm || ||M) that is also a left CG module such that the elements of G act on M as continuous C-linear transformations. We study this group for G an infinite, finitely generated group. Of particular interest are the implications of the vanishing of the group H(G,M). The first result is that H(G,CG) imbeds...