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SVariable -- Specifies symbol representing s-functions

Description

This is an optional argument for the constructor of a Symmetric ring. It indicates the symbol to be used to denote s-functions in the associated Schur ring. The default value is s.

i1 : R = symmetricRing(QQ,5,SVariable => getSymbol"s");
 
i2 : S = schurRing R

o2 = S

o2 : SchurRing
 
i3 : s_2^2

o3 = s  + s    + s
      4    3,1    2,2

o3 : S

The chosen symbol becomes the actual indexing name for the basis elements of the Schur ring. Any symbol may be used; for instance, sigma yields Schur elements sigma_\lambda which multiply by Littlewood-Richardson:

i4 : T = schurRing(QQ,sigma,4)

o4 = T

o4 : SchurRing
 
i5 : sigma_{1,1}^2

o5 = sigma    + sigma      + sigma
          2,2        2,1,1        1,1,1,1

o5 : T

An Sn-flavored ring using a custom SVariable stores characters of the symmetric group on that symbol:

i6 : Sn = schurRing(QQ,sig,4,GroupActing => "Sn");
 
i7 : sig_{3,1} * sig_{2,2}

o7 = sig    + sig
        3,1      2,1,1

o7 : Sn

Distinct SVariable choices allow a tensor product of two Schur rings to carry unambiguous variable names for each factor, so a bi-representation has a clean display:

i8 : S = schurRing(QQ,s,3);
 
i9 : T = schurRing(S,tau,2);
 
i10 : s_{2,1} * tau_{1,1}

o10 = s   tau
       2,1   1,1

o10 : T

See also

Functions with optional argument named SVariable:

  • schurRing(...,SVariable=>...)
  • symmetricRing(...,SVariable=>...)

For the programmer

The object SVariable is a symbol.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SchurRings.m2:7762:0.