toGL -- Express an element in the plain GL Schur basis

Usage:

g = toGL f

g = toGL(f, T)
Inputs:
- f, a ring element, any symmetric-function-like element (in a SchurRing of any flavor or in a symmetricRing)
- T, a Schur ring, optional target ring; must have GroupActing => "GL"
Outputs:
- g, a ring element, f re-expressed in the Schur basis of an associated (or user-supplied) GL character ring

Description

toGL is a readability synonym for toS. It is provided so that calling code documenting an intent to obtain a GL character reads naturally, in contrast to the more neutral toS. Conversions from Basis "Monomial" rings and from GroupActing-variant rings ("Sp", "O", "SL", "RatGL") are handled transparently.

i1 : R = symmetricRing(QQ, 4);

i2 : toGL(R_{0} * R_{5})  -- e_1 * h_2

o2 = s  + 4s    + 5s    + 6s      + 5s      + 4s
      5     4,1     3,2     3,1,1     2,2,1     2,1,1,1

o2 : schurRing (QQ, s, 4)

i3 : T = schurRing(QQ, t, 3);

i4 : toGL(R_{0} * R_{5}, T)

o4 = t  + 4t    + 5t    + 6t      + 5t
      5     4,1     3,2     3,1,1     2,2,1

o4 : T

toGL and toS produce the same output on symmetric-ring elements; only the name of the function differs:

i5 : toGL(e_1 * h_2) == toS(e_1 * h_2)

o5 = true

i6 : toGL(h_3 - p_3) == toS(h_3 - p_3)

o6 = true

A monomial-basis Schur ring expands to the plain Schur basis in the same way as a Kostka-inverse computation: the monomial symmetric function $m_\lambda$ is a signed sum of Schur functions.

i7 : M = schurRing(QQ, m, 3, Basis => "Monomial");

i8 : toGL m_{2,1}

o8 = s    - 2s
      2,1     1,1,1

o8 : schurRing (QQ, s, 3)

Conversion from an Sp-ring applies the Koike-Terada branching rule that expands a symplectic character in the GL Schur basis; the GL(3) Schur expansion of the $Sp(6)$ character $sp_{2,1}$ is:

i9 : Sp = schurRing(QQ, sp, 3, GroupActing => "Sp");

i10 : toGL sp_{2,1}

o10 = s    - s
       2,1    1

o10 : schurRing (QQ, s, 3)

Finally, toGL of a polynomial expression built from $e$-, $h$-, or $p$-generators in a symmetricRing is an efficient way to ask for its Schur decomposition:

i11 : U = symmetricRing(QQ, 4);

i12 : toGL(e_1 * p_2 + h_3)

o12 = 2s  - s
        3    1,1,1

o12 : schurRing (QQ, s, 4)

Ways to use toGL:

toGL(RingElement)
toGL(RingElement,SchurRing)

For the programmer

The object toGL is a method function.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SchurRings.m2:8817:0.

toGL -- Express an element in the plain GL Schur basis

Description

See also

Ways to use toGL:

For the programmer