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Combinatorial Algorithms for Binary Operations on LR-tableaux with Entries Equal to 1 with Applications to Nilpotent Linear Operators

Kaniecki Mariusz, Kosakowska Justyna

Fundamenta Informaticae

2020

Vol. 174, nr 2

121--136

In the paper we investigate an algorithmic associative binary operation * on the set ℒℛ1 of Littlewood-Richardson tableaux with entries equal to one. We extend * to an algorithmic nonassociative binary operation on the set ℒℛ1 × ℕ and show that it is equivalent to the operation of taking the generic extensions of objects in the category of homomorphisms from semisimple nilpotent linear operators to nilpotent linear operators. Thus we get a combinatorial algorithm computing generic extensions in this category.

Between "very large" and "infinite" : The asymptotic representation theory

Vershik A.

Probability and Mathematical Statistics

2013

Vol. 33, Fasc. 2

467--476

I illustrate the historical roots of the theory which I called later “Asymptotic Representation Theory” – the theory which can be considered as a part functional analysis, representation theory, and more general – probability theory, asymptotic combinatorics, the theory of random matrices, dynamics, etc. The first and very concrete example is a remarkable (and forgotten) paper by J. von Neumann, which I try here to connect with the modern theory of random matrices; the second example is a quote of an important thought of H.Weyl about the theory of symmetric groups. In the last section I give a short review of the ideas of the asymptotic representation theory, which was developed starting from the 1970s, and now became very popular. I mention several important problems, and give a list (incomplete) of references. But the reader must remember that this is just a synopsis of the “baby talk”.