jacobiTrudi -- Jacobi-Trudi determinant

Usage:

f = jacobiTrudi(lambda,R)
Inputs:
- lambda, a basic list, 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, a ring element, 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)

Selecting EorH => "H" uses the conjugate determinant formula $s_\lambda = \det(h_{\lambda_i - i + j})$. The two branches produce different e- vs h-polynomials but always represent the same Schur function:

i5 : R = symmetricRing(QQ,8);

i6 : lam = {4,3,2,1};

i7 : fe = jacobiTrudi(lam,R,EorH => "E");

i8 : fh = jacobiTrudi(lam,R,EorH => "H");

i9 : toS fe

o9 = s
      4,3,2,1

o9 : schurRing (QQ, s, 8)

i10 : toS fh

o10 = s
       4,3,2,1

o10 : schurRing (QQ, s, 8)

i11 : toS fe == toS fh

o11 = true

The routine caches intermediate subdeterminants on the ring via jacobiTrudi(...,Memoize=>...), so a second call on a large partition returns almost instantly:

i12 : R = symmetricRing(QQ,6);

i13 : elapsedTime jacobiTrudi({4,3,2,1},R);
 -- .000388909s elapsed

i14 : elapsedTime jacobiTrudi({4,3,2,1},R);
 -- .000012854s elapsed

Passing a partition through toSymm applied to the corresponding Schur label reproduces the Jacobi-Trudi output:

i15 : R = symmetricRing(QQ,5);

i16 : S = schurRing R;

i17 : jacobiTrudi({3,2,1},R) == toSymm(S_{3,2,1})

o17 = true

jacobiTrudi works over tensor products of Symmetric rings, producing a determinant in the outermost set of generators:

i18 : R = symmetricRing(QQ, 4, EHPVariables => (a,b,c));

i19 : T = symmetricRing(R, 3, EHPVariables => (x,y,z));

i20 : jacobiTrudi({2,1},T)

o20 = x x  - x
       1 2    3

o20 : T

i21 : jacobiTrudi({3,2},T, EorH => "H")

o21 = y y
       2 3

o21 : T

Ways to use jacobiTrudi:

jacobiTrudi(BasicList,Ring)

For the programmer

The object jacobiTrudi is a method function with options.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SchurRings.m2:6102:0.