A proof of the second Rogers-Ramanujan identity via Kleshchev multipartitions

We give another proof of the second Rogers-Ramanujan identity by Kashiwara crystals.Comment: 6 pages, a submitted versio

A Fibonacci variant of the Rogers-Ramanujan identities via crystal energy

We define a length function for a perfect crystal. As an application, we derive a variant of the Rogers-Ramanujan identities which involves (a $q$-analog of) the Fibonacci numbers.Comment: This is an early draft for a talk for the conference "Recent developments in Combinatorial Representation Theory" at RIM

A new four parameter q-series identity and its partition implications

We prove a new four parameter q-hypergeometric series identity from which the three parameter key identity for the Goellnitz theorem due to Alladi, Andrews, and Gordon, follows as a special case by setting one of the parameters equal to 0. The new identity is equivalent to a four parameter partition theorem which extends the deep theorem of Goellnitz and thereby settles a problem raised by Andrews thirty years ago. Some consequences including a quadruple product extension of Jacobi's triple product identity, and prospects of future research are briefly discussed.Comment: 25 pages, in Sec. 3 Table 1 is added, discussion is added at the end of Sec. 5, minor stylistic changes, typos eliminated. To appear in Inventiones Mathematica