## Partition identities and $A_n^{(1)}$ crystals

#### Jehanne DousseCNRS, Université de Lyon

There is a rich interaction between the representation theory of affine Lie algebras and partition identities. This connection was first revealed in the 1980's, when Lepowsky and Wilson gave a proof of the Rogers-Ramanujan identities using representations of $A_1^{(1)}$. Other representation theorists have then extended their method and obtained new identities yet unknown to combinatorialists. For example, Capparelli's identity came from representations of $A_2^{(2)}$ and Primc's identities from the crystal base theory of $A_1^{(1)}$ and $A_2^{(1)}$.
We give an infinite family of identities which generalise Primc's identities, and show that these generalisations are related to the crystal base theory of $A_n^{(1)}$ for all n. Through a bijection, we also obtain two infinite families of identities generalising Capparelli's identity. These three families of partition identities relate partitions with difference conditions (coming from the Lie
algebras) to refinements of Frobenius partitions, which are well suited for asymptotic analysis. Moreover, our generalisation of Primc's identities yields a new character formula with obviously positive coefficients for the level 1 standard modules of $A_n^{(1)}$ for all n, and also recovers the Kac-Peterson character formula.
This is joint work with Isaac Konan.

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