Generalisations of Capparelli's and Primc's identities, II: perfect $A_{n-1}^{(1)}$ crystals and explicit character formulas

Abstract: In the first paper of this series, we gave infinite families of coloured partition identities which generalise Primc's and Capparelli's classical identities. In this second paper, we study the representation theoretic consequences of our combinatorial results. First, we show that the difference conditions we defined in our $n^2$-coloured generalisation of Primc's identity, which have a very simple expression, are actually the energy function with values in ${0,1,2}$ for the perfect crystal of the tensor product of the vector representation and its dual in $A_{n-1}^{(1)}$. Then we introduce a new type of partitions, grounded partitions, which allows us to retrieve connections between character formulas and partition generating functions without having to perform a specialisation. Finally, using the formulas for the generating functions of our generalised partitions, we recover the Kac-Peterson character formula for the characters of all the irreducible highest weight $A_{n-1}^{(1)}$-modules of level $1$, and give a new character formula as a sum of infinite products with obviously positive coefficients in the generators $e^{- \alpha_i} \ (i \in {0, \dots , n-1}),$ where the $\alpha_i$'s are the simple roots.