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GroupActing -- Specifies the group that is acting

Description

This is an optional argument for the schurRing and symmetricRing functions. When the exterior or symmetric powers of a symmetric function g are computed, the result depends on whether g is interpreted as a virtual representation of a general linear, symmetric, symplectic, or orthogonal group. The option GroupActing specifies the interpretation to be considered. Its possible values are "GL" (the default), "Sn", "SL", "Sp", "O", and "RatGL".

i1 : S = schurRing(s,2);
 
i2 : exteriorPower(3,s_2)

o2 = s
      3,3

o2 : S
 
i3 : T = schurRing(t,2,GroupActing => "Sn");
 
i4 : symmetricPower(2,t_{1,1})

o4 = t
      2

o4 : T

The first example computes the decomposition of \Lambda^3(Sym^2(V)) into irreducible GL(V)-representations, while the second one computes the second symmetric power of the sign representation of the symmetric group S_2.

Multiplication differs sharply between the "GL" and "Sn" interpretations. Under "GL", the product is the Littlewood-Richardson tensor product of polynomial representations, which is degree-additive in the partitions. Under "Sn", multiplication is the ordinary tensor product of characters of a single symmetric group, so only partitions of the same size may be multiplied, and the product is expanded in the Kronecker coefficients:

i5 : Sgl = schurRing(QQ,s,4);
 
i6 : s_{2,1} * s_{2,1}

o6 = s    + s      + s    + 2s      + s        + s      + s
      4,2    4,1,1    3,3     3,2,1    3,1,1,1    2,2,2    2,2,1,1

o6 : Sgl
 
i7 : Ssn = schurRing(QQ,t,4,GroupActing => "Sn");
 
i8 : t_{2,1,1} * t_{2,1,1}

o8 = t  + t    + t    + t
      4    3,1    2,2    2,1,1

o8 : Ssn

The values "Sp" and "O" select the stable (universal) character ring of the symplectic and orthogonal groups, respectively. The basis elements are indexed by partitions, with sp_\lambda (respectively o_\lambda) standing for the irreducible symplectic (respectively orthogonal) character associated to \lambda. Multiplication in these rings is the Newell-Littlewood product, implemented by conversion to the Schur basis via Koike's branching formulas and back:

i9 : Sp = schurRing(QQ,sp,GroupActing => "Sp");
 
i10 : sp_{1} * sp_{1}

o10 = sp  + sp    + sp
        2     1,1     ()

o10 : Sp
 
i11 : sp_{1,1}

o11 = sp
        1,1

o11 : Sp
 
i12 : toS sp_{1,1}

o12 = s    - s
       1,1    ()

o12 : schurRing (QQ, s, infinity)
 
i13 : O = schurRing(QQ,o,GroupActing => "O");
 
i14 : o_{1} * o_{1}

o14 = o  + o    + o
       2    1,1    ()

o14 : O
 
i15 : toS o_{2}

o15 = s  - s
       2    ()

o15 : schurRing (QQ, s, infinity)

Exterior and symmetric powers are likewise reinterpreted. In the orthogonal ring, exterior powers of the second fundamental character mix several Newell-Littlewood terms:

i16 : Ofin = schurRing(QQ,obar,5,GroupActing => "O");
 
i17 : exteriorPower(3, obar_2)

o17 = obar      + obar    + obar    + obar      + obar
          4,1,1       3,3       3,1       2,1,1       1,1

o17 : Ofin

Setting GroupActing => "SL" forces the top-row reduction s_\lambda = s_{(\lambda_1 - \lambda_n, \ldots, \lambda_{n-1} - \lambda_n)}, reflecting the fact that the determinant representation is trivial in SL. Long partitions collapse accordingly:

i18 : SL3 = schurRing(QQ,sl,3,GroupActing => "SL");
 
i19 : sl_{3,2,1}

o19 = sl
        2,1

o19 : SL3
 
i20 : sl_{4,2,2}

o20 = sl
        2

o20 : SL3
 
i21 : sl_{2,1,1}

o21 = sl
        1

o21 : SL3

Finally, GroupActing => "RatGL" produces the ring of rational (mixed) polynomial representations of GL(V), whose characters are indexed by pairs of partitions (bipartitions), and whose tensor product is again a Littlewood-Richardson-style rule:

i22 : Rg = schurRing(QQ,r,3,GroupActing => "RatGL");
 
i23 : r_({1},{}) * r_({},{1})

o23 = r           + r
       {{1}, {1}}    {{}, {}}

o23 : Rg

Stable rings (no dimension specified) allow arbitrarily many parts in the partition labels; a concrete dimension can be supplied to restrict the partitions and to model a finite-dimensional group.

See also

Functions with optional argument named GroupActing:

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

For the programmer

The object GroupActing is a symbol.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SchurRings.m2:7927:0.