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# jacobiTrudi -- Jacobi-Trudi determinant

## Synopsis

• Usage:
f = jacobiTrudi(lambda,R)
• Inputs:
• lambda, , a nonincreasing list of integers, or a partition
• R, a ring, a Symmetric ring
• Optional inputs:
• EorH => ..., default value "E", e- or h- representation of Jacobi-Trudi determinant
• Memoize => ..., default value true, Store values of the jacobiTrudi function.
• Outputs:
• f, , an element of a Symmetric ring

## Description

Given a partition lambda and Symmetric ring R, the method evaluates the Jacobi-Trudi determinant corresponding to the partition lambda, yielding a representation of the Schur function s_{lambda} as a symmetric function in R. The default option is to represent this symmetric function in terms of e-polynomials.

 i1 : R = symmetricRing(QQ,10); i2 : jacobiTrudi({3,2,2,1},R) 2 o2 = e e e - e - e e e + e e 1 3 4 4 1 2 5 2 6 o2 : R i3 : jacobiTrudi(new Partition from {4,4,1},R,EorH => "H") 2 o3 = h h - h h h - h h + h h 1 4 1 3 5 4 5 3 6 o3 : R i4 : toS oo o4 = s 4,4,1 o4 : schurRing (QQ, s, 10)

## Ways to use jacobiTrudi :

• jacobiTrudi(BasicList,Ring)

## For the programmer

The object jacobiTrudi is .