arXiv subjects

Stefano Capparelli

Publications and source records attributed to Stefano Capparelli.

7 recordsLinked to original sources

arXiv2025

Leading Terms of Relations on a Level 5 Module over the Twisted Affine Lie Algebra $A_2^{(2)}$

One of the starting points of this work was the duality of Borcea relating standard level $k$ representations of $A_1^{(1)}$ and level $2k+1$ of $A_2^{(2)}$. For $k=1$, the combinatorial bases in both cases yield the two Capparelli identities and we wanted to see if there is a correspondence between the bases in terms of partitions for all $k\in\mathbb N$. By using the vertex operator relations in the principal picture for level $5$ standard $A_2^{(2)}$-modules, we reduce a spanning set of Poincar\'e-Birkhoff-Witt-type vectors in $L(5\Lambda_0)$ by removing the leading terms of relations and rendering a list of 34 ''difference'' conditions for partitions. Using computer programs, we enumerated the partitions satisfying these conditions and obtained a truncated generating series agreeing with the principally specialized character for all powers of $q$ up to $41$. Although our list of leading terms is incomplete, our results show that the corresponding combinatorial identity for $L_{A_2^{(2)}}(5\Lambda_0)$ drastically differs from the one for the Borcea dual $L_{A_1^{(1)}}(2\Lambda_0)$.

math.CO↗

arXiv2020

Some partition and analytical identities arising from Alladi, Andrews, Gordon bijections

In a work of 1995, Alladi, Andrews, and Gordon provided a generalization of the two Capparelli identities involving certain classes of integer partitions. Inspired by that contribution, in particular as regards the general setting and the tools the authors employed, we obtain new partition identities by identifying further sets of partitions that can be explicitly put into a one-to-one correspondence by the method described in the 1995 paper. As a further result, although of a different nature, we obtain an analytical identity of Rogers-Ramanujan type, involving generating functions, for a class of partition identities already found in that paper and that generalize the first Capparelli identity and include it as a particular case. To achieve this, we apply the same strategy as Kanade and Russel did in a recent paper. This method relies on the use of jagged partitions that can be seen as a more general kind of integer partitions.

math.NT↗

arXiv2018

Chebyshev coordinates and Salem numbers

By expressing polynomials in the basis of Chebyshev polynomials, certain families of hyperbolic polynomials appear naturally. Some of these families have all their roots in the interval $[-2,2]$. In many cases the span of the family of polynomials thus found is greater than 4, and we show that they are the minimal polynomials of Salem numbers, possibly multiplied by some cyclotomic polynomials. In addition, we show how to compute the limit of the largest and smallest roots.

math.CO↗

arXiv2015

A non-existence result on symplectic semifield spreads

We prove that there do not exist non-Desarguesian symplectic semifield spreads of PG$(5,q^2)$, $q\geq 2^{14}$ even, whose associated semifield has center containing $\mathbb{F}_q$, by proving that the only $\mathbb{F}_q$-linear set of rank 6 disjoint from the secant variety of the quadric Veronese variety of PG$(5,q^2)$ is a plane with three points of the Veronese surface of PG$(5,q^6)\setminus$PG$(5,q^2)$.

math.CO↗

arXiv2015

Some results on Dyck paths and Motzkin paths

We introduce an equivalence relation on the set of Dyck paths and some operations on them. We determine a formula for the cardinality of those equivalence classes and use this information to obtain a combinatorial formula for the number of Dyck and Motzkin paths of a fixed length.

math.CO↗

arXiv2006

The Rogers-Selberg recursions, the Gordon-Andrews identities and intertwining operators

Using the theory of intertwining operators for vertex operator algebras we show that the graded dimensions of the principal subspaces associated to the standard modules for $\hat{\goth{sl}(2)}$ satisfy certain classical recursion formulas of Rogers and Selberg. These recursions were exploited by Andrews in connection with Gordon's generalization of the Rogers--Ramanujan identities and with Andrews' related identities. The present work generalizes the authors' previous work on intertwining operators and the Rogers--Ramanujan recursion.

math.QA↗

arXiv2004

The Rogers--Ramanujan recursion and intertwining operators

We use vertex operator algebras and intertwining operators to study certain substructures of standard $A_1^{(1)}$--modules, allowing us to conceptually obtain the classical Rogers--Ramanujan recursion. As a consequence we recover Feigin-Stoyanovsky's character formulas for the principal subspaces of the level 1 standard $A_1^{(1)}$--modules.

math.QA↗