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• W2989631839 endingPage "108571" @default.
• W2989631839 startingPage "108571" @default.
• W2989631839 abstract "The partition identities of Capparelli and Primc were originally discovered via representation theoretic techniques, and have since then been studied and refined combinatorially, but the question of giving a very broad generalisation remained open. In these two companion papers, we give infinite families of partition identities which generalise Primc's and Capparelli's identities, and study their consequences on the theory of crystal bases of the affine Lie algebra An−1(1). In this first paper, we focus on combinatorial aspects. We give a n2-coloured generalisation of Primc's identity by constructing a n2×n2 matrix of difference conditions, Primc's original identities corresponding to n=2 and n=3. While most coloured partition identities in the literature connect partitions with difference conditions to partitions with congruence conditions, in our case, the natural way to generalise these identities is to relate partitions with difference conditions to coloured Frobenius partitions. This gives a very simple expression for the generating function. With a particular specialisation of the colour variables, our generalisation also yields a partition identity with congruence conditions. Then, using a bijection from our new generalisation of Primc's identity, we deduce a large family of identities on (n2−1)-coloured partitions which generalise Capparelli's identity, also in terms of coloured Frobenius partitions. The particular case n=2 is Capparelli's identity and one of the cases where n=3 recovers an identity of Meurman and Primc. In the second paper, we will focus on crystal theoretic aspects. We will show that the difference conditions we defined in our n2-coloured generalisation of Primc's identity are actually energy functions for certain An−1(1) crystals. We will then use this result to retrieve the Kac-Peterson character formula and derive a new character formula as a sum of infinite products for all the irreducible highest weight An−1(1)-modules of level 1." @default.
• W2989631839 created "2019-12-05" @default.
• W2989631839 creator A5028754105 @default.
• W2989631839 creator A5057691546 @default.
• W2989631839 date "2022-10-01" @default.
• W2989631839 modified "2023-10-14" @default.
• W2989631839 title "Generalisations of Capparelli's and Primc's identities, I: Coloured Frobenius partitions and combinatorial proofs" @default.
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• W2989631839 doi "https://doi.org/10.1016/j.aim.2022.108571" @default.
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