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toS -- Schur (s-) basis representation

Description

Given a symmetric function f, the function toS yields a representation of f as a linear combination of Schur functions.

If f is an element of a Symmetric ring R and the output Schur ring S is not specified, then the output fs is an element of the Schur ring associated to R (see schurRing).

i1 : R = symmetricRing(QQ,4);
 
i2 : fs = toS(e_1*h_2+p_3)

o2 = 2s  + s
       3    1,1,1

o2 : schurRing (QQ, s, 4)
 
i3 : S = schurRing(s,2);
 
i4 : toS(fs,S)

o4 = 2s
       3

o4 : S

This also works over tensor products of Symmetric/Schur rings.

i5 : R = symmetricRing(4, EHPVariables => (a,b,c), SVariable => r);
 
i6 : S = symmetricRing(R, 2, EHPVariables => (x,y,z), SVariable => s);
 
i7 : T = symmetricRing(S, 3, SVariable => t);
 
i8 : A = schurRing T;
 
i9 : f = a_3*x_2*e_1 - b_1*z_2*p_3

o9 = - b z p  + a x e
        1 2 3    3 2 1

o9 : T
 
i10 : toS f

o10 = (- r s  + r s   )t  + (r s  - r s   )t    + (- r s  + r s   )t      +
          1 2    1 1,1  3     1 2    1 1,1  2,1       1 2    1 1,1  1,1,1  
      -----------------------------------------------------------------------
      r     s   t
       1,1,1 1,1 1

o10 : A

The Jacobi-Trudi determinant $s_\lambda = \det(h_{\lambda_i - i + j})$ is inverted by toS: feeding a Jacobi-Trudi expression back through toS recovers a single Schur label.

i11 : R = symmetricRing(QQ,5);
 
i12 : toS jacobiTrudi({3,2,1},R)

o12 = s
       3,2,1

o12 : schurRing (QQ, s, 5)

GL to Sn. The GL Schur basis and the Frobenius-characteristic (Sn) basis share the same partition index set. Combining toS with toSn carries coefficient data between the two flavors.

i13 : G  = schurRing(QQ, g, 4);
 
i14 : Sn = schurRing(QQ, n, 4, GroupActing => "Sn");
 
i15 : a  = toSn(g_{2,1}, Sn)

o15 = n
       2,1

o15 : Sn
 
i16 : toS(a, G)

o16 = g
       2,1

o16 : G

Sp {\rm to} S {\rm to} Sp. Elements of a variant-basis ring (e.g.\ "Sp", "O") can be expanded into the plain GL Schur basis with toS, and the resulting combination re-expressed in a variant-basis ring via convert.

i17 : Sp = schurRing(QQ, sp, 4, GroupActing => "Sp");
 
i18 : f  = toS sp_{2,1}

o18 = s    - s
       2,1    1

o18 : schurRing (QQ, s, 4)
 
i19 : Sp' = schurRing(QQ, sq, 4, GroupActing => "Sp");
 
i20 : convert(f, Sp')

o20 = sq
        2,1

o20 : Sp'

See also

Ways to use toS:

  • toS(RingElement)
  • toS(RingElement,SchurRing)

For the programmer

The object toS is a method function.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SchurRings.m2:5727:0.