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On strongly sum-free subsets of abelian groups

Tomasz Łuczak, Tomasz Schoen (1996)

Colloquium Mathematicae

In his book on unsolved problems in number theory [1] R. K. Guy asks whether for every natural l there exists n 0 = n 0 ( l ) with the following property: for every n ≥ n 0 and any n elements a 1 , . . . , a n of a group such that the product of any two of them is different from the unit element of the group, there exist l of the a i such that a i j a i k ≠ a m for 1 ≤ j < k ≤ l and 1 ≤ m ≤ n . In this note we answer this question in the affirmative in the first non-trivial case when l=3 and the group is abelian, proving the following result.

Radicals of symmetric cellular algebras

Yanbo Li (2013)

Colloquium Mathematicae

For a symmetric cellular algebra, we study properties of the dual basis of a cellular basis first. Then a nilpotent ideal is constructed. The ideal connects the radicals of cell modules with the radical of the algebra. It also yields some information on the dimensions of simple modules. As a by-product, we obtain some equivalent conditions for a finite-dimensional symmetric cellular algebra to be semisimple.

RSK bases and Kazhdan-Lusztig cells

K. N. Raghavan, Preena Samuel, K. V. Subrahmanyam (2012)

Annales de l’institut Fourier

From the combinatorial characterizations of the right, left, and two-sided Kazhdan-Lusztig cells of the symmetric group, “ RSK bases” are constructed for certain quotients by two-sided ideals of the group ring and the Hecke algebra. Applications to invariant theory, over various base rings, of the general linear group and representation theory, both ordinary and modular, of the symmetric group are discussed.

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