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The Rogers-Ramanujan identities: Lie theoretic interpretation and proof
1Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903.
Summary
This study introduces an abstract Rogers-Ramanujan identity for Kac-Moody Lie algebra modules. It connects classical identities to characters of standard modules, generalizing known results.
Area of Science:
- Representation Theory
- Lie Algebras
- Number Theory
Background:
- The classical Rogers-Ramanujan identities relate infinite products to infinite sums.
- These identities are linked to characters of standard modules for the Euclidean Kac-Moody Lie algebra A(1)((1)).
- A Heisenberg subalgebra of A(1)((1)) has been constructed using differential operators and exponential generating functions.
Purpose of the Study:
- To formulate a natural "abstract Rogers-Ramanujan identity" applicable to arbitrary standard A(1)((1))-modules.
- To demonstrate that this abstract identity specializes to the classical Rogers-Ramanujan identities for specific standard modules.
- To explore the connection between the abstract identity, characters of modules, and the dimensions of spaces of symmetric polynomials.
Main Methods:
- Utilizing a Heisenberg subalgebra ([unk]) of the Kac-Moody Lie algebra A(1)((1)).
- Formulating an abstract identity for the principally specialized character of the space of highest weight vectors (vacuum states) for [unk].
- Realizing this space as the span of symmetric polynomials derived from exponential generating functions.
Main Results:
- An abstract Rogers-Ramanujan identity is established for general standard A(1)((1))-modules.
- The abstract identity equates a product and a sum for the principally specialized character of the vacuum state space.
- The summands in the abstract identity correspond to the dimensions of spaces of symmetric polynomials.
Conclusions:
- The abstract identity provides a unified framework for understanding Rogers-Ramanujan type identities within the context of Lie algebra representation theory.
- The study conjectures that these abstract identities generalize the Rogers-Ramanujan identities further, aligning with work by Gordon, Andrews, and Bressoud.
- This research deepens the connection between formal power series identities, Lie algebra characters, and combinatorial structures like symmetric polynomials.