Algebra & Discrete Mathematics
|Speaker:||Prof. Ernesto Vallejo, Instituto de Matematicas, UNAM, Mexico|
|Start time:||Thu, May 23 2002, 3:10PM|
A formula, due to Snapper, gives the number of 3-dimensional (0,1)-matrices with fixed plane sums as an inner product of certain characters of the symmetric group. Using this formula we give a criterion, in the spirit of the Gale-Ryser theorem, for deciding when such number is one. In doing this we introduce the notion of "minimal" matrix. Snapper's formula also permits us to establish a link between 3-dimensional matrices and the problem of determining the minimal components, in the dominance order, of the Kronecker product of two irreducible characters of the symmetric group. For this, it is useful to know the complete list of minimal matrices with prescribed row and column sum vectors. We illustrate this by showing how the classification of minimal matrices of size 2 by q gives information on the minimal components of products of some irreducible characters.