On partition identities of Capparelli and Primc

Dousse, Jehanne

doi:10.48550/arxiv.1811.02251

Cited by 3 publications

(13 citation statements)

References 16 publications

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“…Both Capparelli's identity and Meurman and Primc's identity with difference conditions (1.13) did not have any apparent connection to the theory of perfect crystals. The bijection between P 2 and CC 2 in [Dou18b] gave an unexpected connection with Primc's identity and the theory of perfect crystals. The present theorem shows that Meurman and Primc's identity with difference conditions (1.13) can actually be deduced from Primc's Theorem 1.6.…”

Section: Statement Of Resultsmentioning

confidence: 97%

“…The principle of the method of weighted words, introduced by Alladi and Gordon to refine Schur's identity [AG93], is to give an identity on coloured partitions, which under certain transformations on the coloured partitions, becomes the original identity. We now describe Alladi, Andrews, and Gordon's refinement of Capparelli's identity (slightly reformulated by the first author in [Dou18b]).…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…But the theorem above shows that keeping track of all colours except b leads to a much more intricate infinite product as well, and that the extremely simple expression 1/(q; q) ∞ appears only because of the particular dilation that Primc considered. Later, the first author [Dou18b] even gave an expression for the generating function for P (n; i, j, k, ) keeping track of all the colours, but it can be written as an infinite product only if we do not keep track of the colour b.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…1 , Capparelli's and Primc's identities didn't seem related from the representation theoretic point of view, as they were obtained in completely different ways, and Capparelli's identity did not seem related to perfect crystals. However, recently, the first author [Dou18b] gave a bijection between coloured partitions satisfying the difference conditions (1.4) and pairs of partitions (λ, µ), where λ is a coloured partition satisfying the difference conditions (1.3), and µ is a partition coloured b. This bijection preserves the total weight, the number of parts, the size of the parts, and the number of parts coloured a and d. Therefore, combinatorially, these two identities are very closely related.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…Recently in [Dou18b], the first author gave a bijection between Primc's partitions P 2 and pairs (λ, µ) where λ ∈ C 2 is a Capparelli partition and µ is a classical partition. This bijection only modifies some free colours, so it preserves the weight, the number of parts, the size of the parts, and the number of appearances of colours a and d. In this way, she showed that Capparelli's identity is very closely related to Primc's identity and can be deduced from it, even though until then, these two identities seemed unrelated from the representation theoretic point of view.…”

Section: Statement Of Resultsmentioning

confidence: 99%

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Generalisations of Capparelli's and Primc's identities, I: coloured Frobenius partitions and combinatorial proofs

Dousse¹,

Konan²

2019

Preprint

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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 A(1)In this first paper, we focus on combinatorial aspects. We give a n 2 -coloured generalisation of Primc's identity by constructing a n 2 × n 2 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 two families of identities on (n 2 − 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 the case 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 n 2 -coloured generalisation of Primc's identity are actually energy functions for certain A(1) n−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 A(1) n−1 -modules of level 1.

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Section: Statement Of Resultsmentioning

confidence: 97%

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Section: Statement Of Resultsmentioning

confidence: 99%

See 3 more Smart Citations

Generalisations of Capparelli's and Primc's identities, I: coloured Frobenius partitions and combinatorial proofs

Dousse¹,

Konan²

2019

Preprint

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Sequentially Congruent Partitions and Related Bijections

Schneider

2019

Ann. Comb.

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In recent work, M. Schneider and the first author studied a curious class of integer partitions called "sequentially congruent" partitions: the mth part is congruent to the (m + 1)th part modulo m, with the smallest part congruent to zero modulo the number of parts. Let p S (n) be the number of sequentially congruent partitions of n, and let p (n) be the number of partitions of n wherein all parts are squares. In this note we prove bijectively, for all n ≥ 1, that p S (n) = p (n). Our proof naturally extends to show other exotic classes of partitions of n are in bijection with certain partitions of n into kth powers.

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qFunctions -- A Mathematica package for $q$-series and partition theory applications

Ablinger¹,

Uncu²

2019

Preprint

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We describe the qFunctions Mathematica package for q-series and partition theory applications. This package includes both experimental and symbolic tools. The experimental set of elements includes guessers for q-shift equations and recurrences for given q-series and fitting/finding explicit expressions for sequences of polynomials. This package can symbolically handle formal manipulations on q-differential, q-shift equations and recurrences, such as switching between these forms, finding the greatest common divisor of recurrences, and formal substitutions. Here, we also extend the classical method of the weighted words approach. Moreover, qFunctions has implementations that automate the recurrence system creation of the weighted words approach as well as a scheme on cylindric partitions.

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On partition identities of Capparelli and Primc

Cited by 3 publications

References 16 publications

Generalisations of Capparelli's and Primc's identities, I: coloured Frobenius partitions and combinatorial proofs

Generalisations of Capparelli's and Primc's identities, I: coloured Frobenius partitions and combinatorial proofs

Sequentially Congruent Partitions and Related Bijections

qFunctions -- A Mathematica package for $q$-series and partition theory applications

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