La prochaine séance du séminaire Combi du Plateau de Saclay aura lieu ce mercredi à 10h30 dans la salle Philippe Flajolet du LIX. Isaac Konan (IRIF, Université Paris Diderot) nous parlera de “A round trip from crystal bases to integer partitions”.

Le programme du séminaire est disponible ici : https://galac.lri.fr/pages/combi-seminar.html

Résumé: The representation theory of Lie algebras occurs as a rich source of partition identities. This started with Lepowsky and Wilson’s proof of Rogers-Ramanujan identities via the representation of level 3 standard module for the affine type A₁⁽¹⁾. A good example of an identity generator is the (KMN)² crystal base character formula. Starting from this result, Primc gave two identities on the character of the standard module L(Λ₀) for the affine types A₁⁽¹⁾ and A₂⁽¹⁾. In recent works with Dousse, our starting points are the identities given by Primc. We first built a family of colored partitions with n² colors, whose cases n = 2 and n = 3 respectively give the Primc partitions for affine A₁⁽¹⁾ and A₂⁽¹⁾. We then related those Primc generalized partitions to another family of combinatorial objects, the generalized Frobenius partitions, and were able to give nice explicit formulas for their generating functions. Then, we moved back to crystal base theory by proving that our Primc generalized partitions were the corresponding objects for the affine A_n − 1⁽¹⁾. This allowed us to give an explicit non-dilated formula of the characters, not only for L(Λ₀) but for all the level 1 standard modules L(Λ_l) with l ∈ {0, ⋯, n − 1}, as a sum of (n − 1)! series with obvious positive coefficients.