# A.-A. A. Jucys's research while affiliated with Mokslininkų Sąjungos Institutas and other places

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## Publications (3)

It is shown that the number ln of all distinct Latin squares of the nth order appears as a structure constant of the algebra defined on the Magic squares of the same order. The algebra is isomorphic to the algebra of double cosets of the symmetric group of degree n2 with respect to the intransitive subgroup of all substitutions in the n sets of tra...

The homomorphism of a special kind between the ring of symmetric polynomials and the center of the symmetric group ring is established. In the homomorphic mapping of the first ring on the second one the proper values of the images are the values of the corresponding symmetric polynomials with the variables substituted by the set of integers found f...

## Citations

... This turns the calculation of P C ({e α,α }) quite hard, in fact we were not able to find an analytical closed form for it. This problem is similar to those appearing in the enumeration of contingency tables (whose most celebrated examples are the latin and magic squares) and represents still an open problem in combinatorics444546. It is still possible to numerically determine the sum with a computational time growing as M C 2 [ ...

Reference: Combinatorial approach to Modularity

... The term X S is a sum of Jucys-Murphy elements in the algebra CS n [19,31]: it is central in CS n and proportional to identity on any simple CS n -module L (µ) with the coefficient c(µ) (a possible way to check this is to apply (3.40) to M (µ) n ∼ = L (µ) taking into account that d ij M (µ) n = 0 for µ n, see [24]). ...

... per(A) k which has been rearranged and a problem corrected – the last equation in [49] should have f n(n−r)+k instead of f n(n−r) . Jucys [71] constructed an algebra A n over C, with the " magic squares " as a basis, which were actually n × n non-negative integer matrices with row and column sums equal to n. Multiplication in A n was defined using a " structure constant, " which, in one case, was L n . An isomorphism was identified between A n and a subalgebra of the group algebra of the symmetric group S n 2 over C. Representation theory was then used to give an expression for L n in terms of eigenvalues of a particular element of A n . ...