A Probabilistic Proof of the Rogers Ramanujan Identities

The asymptotic probability theory of conjugacy classes of the finite general linear and unitary groups leads to a probability measure on the set of all partitions of natural numbers. A simple method of understanding these measures in terms of Markov chains is given and compared with work on the uniform measure. Elementary probabilistic proofs of the Rogers-Ramanujan identities follow. As a corollary, the main case of Bailey's lemma is interpreted as finding eigenvectors of the transition matrix of the Markov chain. It is shown that the viewpoint of Markov chains extends to quivers.Comment: Final version, to appear in Bull LMS. The one math change is to fix a typo in the limit in Corollary 2. We also make two historical correction

A Trinomial Analogue of Bailey's Lemma and N=2 Superconformal Invariance

We propose and prove a trinomial version of the celebrated Bailey's lemma. As an application we obtain new fermionic representations for characters of some unitary as well as nonunitary models of N = 2 superconformal field theory (SCFT). We also establish interesting relations between N = 1 and N = 2 models of SCFT with central charges $(3/2)( 1 -{2(2 - 4\nu)^2}/{2(4\nu)})$ and $3(1 - 2/{4\nu})$. A number of new mock theta function identities are derived.Comment: Reference and note adde

Polynomial Identities, Indices, and Duality for the N=1 Superconformal Model SM(2,4\nu)

We prove polynomial identities for the N=1 superconformal model SM(2,4\nu) which generalize and extend the known Fermi/Bose character identities. Our proof uses the q-trinomial coefficients of Andrews and Baxter on the bosonic side and a recently introduced very general method of producing recursion relations for q-series on the fermionic side. We use these polynomials to demonstrate a dual relation under q \rightarrow q^{-1} between SM(2,4\nu) and M(2\nu-1,4\nu). We also introduce a generalization of the Witten index which is expressible in terms of the Rogers false theta functions.Comment: 41 pages, harvmac, no figures; new identities, proofs and comments added; misprints eliminate

Generalizations of the Andrews-Bressoud Identities for the N=1N=1 Superconformal Model SM(2,4ν)SM(2,4\nu)

We present generalized Rogers-Ramanujan identities which relate the fermi and bose forms of all the characters of the superconformal model $SM(2,4\nu).$ In particular we show that to each bosonic form of the character there is an infinite family of distinct fermionic $q-$ series representations.Comment: 17 pages in harvmac, no figures, submitted as part of the proceedings of ``Physique et Combinatiore'' held at CIRM Luminy March 27-31, 199

ALMOST A CENTURY OF ANSWERING THE QUESTION: WHAT IS A MOCK THETA FUNCTION?

Quite a few famous and extraordinarily gifted mathematicians led lives that were tragically cut short. Ramanujan is certainly among them. While suffering from a fatal disease, he discovered what he called mock theta functions. Three months before his death in 1920 at the age of 32, he described them in a letter to Hardy that was written under difficultie