Combinatorial Methods for Computing: Plethysms of Schur Functions.
Abstract
The plethysm of two Schur functions s _lambda[ s_mu] was first introduced by Littlewood (13). Littlewood showed that for any partition lambda of m and mu of n, s_ lambda[ s_mu] = sum_nu c_sp{lambda, mu}{nu}s_nu where the sum runs over all partitions nu of mn and c_sp{lambda, mu}{nu} are nonnegative integers. The problem of computing the coefficients c _sp{lambda,mu}{nu } is one of the fundamental open problems in the theory of symmetric functions. In this thesis, we focus on the problem of computing the plethysms s_2[ s_ mu] and s_{1 ^2}[ s_mu] . The problem of computing the Schur function expansion of s_2[ s_{n} ] and s_{1^2 }[ s_{n}] was solved by Littlewood. Recently, Carbonara, Remmel, and Yang (3) gave explicit formulas for the Schur function expansion of the plethysms s_2[ s_mu] and s_ {1^2}[ s_mu] where mu is a hook shape, i.e., mu is of the form (1 ^{k},l) where 1 <=q l. Building on the ideas of Carbonara, Remmel, and Yang we show that one can develop efficient algorithms for computing the Schur function expansion of the plethysms s_2[ s_mu] and s_{1^2}[ s_mu], where mu is of the form (1^{r},n ^{k}) where 1 <=q n, (n^{r},s) where n<=q s,(r,n^{k}), where r<=q n or ((n1)^ {r},n^{k}). The significance of the shapes mu of the form (1 ^{r},n^{k}) where 1<=q n,(n^{r},s) where n<=q s,(r,n^{k}) where r<=q n, or ((n1)^ {r},n^{k}) is that these are a complete list of all shapes mu such that p_2[ s_mu ] is multiplicity free, i.e., those shapes for which the coefficients < p_2 [ s_mu], s_lambda >in{0,+/ 1} . As an application of our algorithms, we derive explicit formulas for the Schur function expansion of the plethysms s_2[ s_mu ] and s_{1^2 }[ s_mu] where mu has either two rows or two columns.
 Publication:

Ph.D. Thesis
 Pub Date:
 January 1995
 Bibcode:
 1995PhDT........48C
 Keywords:

 Mathematics; Physics: Atomic; Physics: Nuclear