New infinite hierarchies of polynomial identities related to the Capparelli partition theorems

Elementary Polynomial Identities Involving $$\varvec{q}$$-Trinomial Coefficients

Annals of Combinatorics ◽

10.1007/s00026-019-00445-8 ◽

2019 ◽

Vol 23 (3-4) ◽

pp. 549-560 ◽

Cited By ~ 1

Author(s):

Alexander Berkovich ◽

Ali Kemal Uncu

Keyword(s):

Polynomial Identities ◽

Binomial Theorem ◽

Infinite Hierarchy ◽

Summation Formulas ◽

Partition Theorems

Abstract We use the q-binomial theorem to prove three new polynomial identities involving q-trinomial coefficients. We then use summation formulas for the q-trinomial coefficients to convert our identities into another set of three polynomial identities, which imply Capparelli’s partition theorems when the degree of the polynomial tends to infinity. This way we also obtain an interesting new result for the sum of the Capparelli’s products. We finish this paper by proposing an infinite hierarchy of polynomial identities.

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Polynomial identities implying Capparelli's partition theorems

Journal of Number Theory ◽

10.1016/j.jnt.2019.02.028 ◽

2019 ◽

Vol 201 ◽

pp. 77-107 ◽

Cited By ~ 1

Author(s):

Alexander Berkovich ◽

Ali Kemal Uncu

Keyword(s):

Polynomial Identities ◽

Partition Theorems

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On Polynomial Identities of Jordan Pairs of Rectangular Matrices

Linear Algebra and its Applications ◽

10.1016/s0024-3795(96)00309-6 ◽

1997 ◽

Vol 260 (1-3) ◽

pp. 257-271 ◽

Cited By ~ 1

Author(s):

A Iltyakov

Keyword(s):

Polynomial Identities

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Polynomial identities related to special Schubert varieties

Applicable Algebra in Engineering Communication and Computing ◽

10.1007/s00200-021-00496-6 ◽

2021 ◽

Author(s):

Francesca Cioffi ◽

Davide Franco ◽

Carmine Sessa

Keyword(s):

Arbitrary Number ◽

Schubert Variety ◽

Decomposition Theorem ◽

Point Of View ◽

Explicit Description ◽

Intersection Cohomology ◽

Schubert Varieties ◽

Polynomial Identities ◽

Poincaré Polynomial

AbstractLet $$\mathcal S$$ S be a single condition Schubert variety with an arbitrary number of strata. Recently, an explicit description of the summands involved in the decomposition theorem applied to such a variety has been obtained in a paper of the second author. Starting from this result, we provide an explicit description of the Poincaré polynomial of the intersection cohomology of $$\mathcal S$$ S by means of the Poincaré polynomials of its strata, obtaining interesting polynomial identities relating Poincaré polynomials of several Grassmannians, both by a local and by a global point of view. We also present a symbolic study of a particular case of these identities.

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