On sums of figurate numbers by using techniques of poset representation theory
Agustin Moreno Ca˜ nadas
Matematicas
UNIVERSIDAD NACIONAL DE COLOMBIA Bogota
Colombia
E-mail address: amorenoca@unal.edu.co
This talk is dedicated to the applications of the poset representation theory to the number theory. We recall that the representation theory of posets was developed in the 70’s by Nazarova and Roiter. The main aim of this study was to determine indecomposable matrix representations of a given poset over a fixed field. Soon after the discovery of matrix representations of a poset, Gabriel introduced the concept of a filtered linear representation of a poset in connection with the investigation of oriented graphs having finitely many isomorphism classes of indecomposable linear representations [3].
In this talk we shall describe how we can use differentiation algorithms (which have been successfully applied in determining the representation type (finite or infinite) of posets and in the classification of indecomposable poset representations), as well as representations over the set of natural numbers of a poset, in order to obtain solutions of some open problems concerning figurate numbers [1,2]. In particular, we present criteria for natural numbers which are the sum of three octahedral numbers (in connection with the Pollock’s conjecture on tetrahedral and octahedral numbers), three polygonal numbers of positive rank (note for example that the problem of representing a natural number as sums of three nonvanishing squares is still an unsolved problem [1]) or four cubes with two of them equal. Some identities of the Rogers- Ramanujan type involving this class of numbers are also obtained with the help of poset representation theory.
References
[1] E. Grosswald, representations of integers as sums of squares, Springer-Verlag, New York, 1985.
[2] R. Guy, Unsolved Problems in Number Theory, 3rd ed., Springer-Verlag, New York, 2004.
[3] D. Simson, Linear Representations of Partially Ordered Sets and Vector Space Categories, Gordon and Breach, London, 1992.
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