4,885 research outputs found
Generalized Irreducible Divisor Graphs
In 1988, I. Beck introduced the notion of a zero-divisor graph of a
commutative rings with . There have been several generalizations in recent
years. In particular, in 2007 J. Coykendall and J. Maney developed the
irreducible divisor graph. Much work has been done on generalized
factorization, especially -factorization. The goal of this paper is to
synthesize the notions of -factorization and irreducible divisor graphs
in domains. We will define a -irreducible divisor graph for non-zero
non-unit elements of a domain. We show that by studying -irreducible
divisor graphs, we find equivalent characterizations of several finite
-factorization properties.Comment: 17 pages, 2 figures, to appear in Communications in Algebr
Higher melonic theories
We classify a large set of melonic theories with arbitrary -fold
interactions, demonstrating that the interaction vertices exhibit a range of
symmetries, always of the form for some , which may be .
The number of different theories proliferates quickly as increases above
and is related to the problem of counting one-factorizations of complete
graphs. The symmetries of the interaction vertex lead to an effective
interaction strength that enters into the Schwinger-Dyson equation for the
two-point function as well as the kernel used for constructing higher-point
functions.Comment: 43 pages, 12 figure
On 1-factorizations of Bipartite Kneser Graphs
It is a challenging open problem to construct an explicit 1-factorization of
the bipartite Kneser graph , which contains as vertices all -element
and -element subsets of and an edge between any
two vertices when one is a subset of the other. In this paper, we propose a new
framework for designing such 1-factorizations, by which we solve a nontrivial
case where and is an odd prime power. We also revisit two classic
constructions for the case --- the \emph{lexical factorization} and
\emph{modular factorization}. We provide their simplified definitions and study
their inner structures. As a result, an optimal algorithm is designed for
computing the lexical factorizations. (An analogous algorithm for the modular
factorization is trivial.)Comment: We design the first explicit 1-factorization of H(2,q), where q is a
odd prime powe
Crystal approach to affine Schubert calculus
We apply crystal theory to affine Schubert calculus, Gromov-Witten invariants
for the complete flag manifold, and the positroid stratification of the
positive Grassmannian. We introduce operators on decompositions of elements in
the type- affine Weyl group and produce a crystal reflecting the internal
structure of the generalized Young modules whose Frobenius image is represented
by stable Schubert polynomials. We apply the crystal framework to products of a
Schur function with a -Schur function, consequently proving that a subclass
of 3-point Gromov-Witten invariants of complete flag varieties for enumerate the highest weight elements under these operators. Included in
this class are the Schubert structure constants in the (quantum) product of a
Schubert polynomial with a Schur function for all . Another by-product gives a highest weight formulation for various fusion
coefficients of the Verlinde algebra and for the Schubert decomposition of
certain positroid classes.Comment: 42 pages; version to appear in IMR
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