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symmetricRing -- Make a Symmetric ring

Description

The method symmetricRing creates a Symmetric ring of dimension n over a base ring A. This is the subring of the ring of symmetric functions over the base A consisting of polynomials in the first n elementary (or complete, or power sum) symmetric functions. If A is not specified, then it is assumed to be QQ.

i1 : R = symmetricRing(QQ[x,y,z],4)

o1 = R

o1 : PolynomialRing
 
i2 : e_2*x+y*p_3+h_2

o2 = y*p  + x*e  + h
        3      2    2

o2 : R
 
i3 : toS oo

o3 = y*s  - y*s    + s  + y*s      + x*s
        3      2,1    2      1,1,1      1,1

o3 : schurRing (QQ[x..z], s, 4)

The elements of a Symmetric ring can be interpreted as characters of either symmetric or general linear groups. This is controlled by the value of the option GroupActing, whose default value is "GL" (general linear group). The other possibility for its value is "Sn" (symmetric group).

i4 : R = symmetricRing(QQ,3,GroupActing => "Sn")

o4 = R

o4 : PolynomialRing
 
i5 : toE symmetricPower(2,e_2)

      2
o5 = e  - e
      1    2

o5 : R

The three symmetric-function generators -- elementary, complete, and power sum -- are all accessible in the same ring, and conversions between them are handled by toE, toH, toP, and toS.

i6 : R = symmetricRing(QQ,5);
 
i7 : toS ((R.pVariable 2)^3)

o7 = s  - s    + 3s    - 2s      - 3s    + 2s        + 3s      - 3s        +
      6    5,1     4,2     4,1,1     3,3     3,1,1,1     2,2,2     2,2,1,1  
     ------------------------------------------------------------------------
     s
      2,1,1,1,1

o7 : schurRing (QQ, s, 5)
 
i8 : toH (R.pVariable 2)

        2
o8 = - h  + 2h
        1     2

o8 : R
 
i9 : toS (R.eVariable 2 * R.hVariable 2)

o9 = s    + s
      3,1    2,1,1

o9 : schurRing (QQ, s, 5)

A Symmetric ring can be built over any commutative base; coefficients can be polynomial rings, modular rings, or even other Schur rings.

i10 : R7 = symmetricRing(ZZ/7,4);
 
i11 : (R7.eVariable 2)^3

       3
o11 = e
       2

o11 : R7
 
i12 : Rt = symmetricRing(QQ[t],3,GroupActing => "Sn");
 
i13 : toS(Rt.eVariable 2 * t)

o13 = t*s
         1,1

o13 : schurRing (QQ[t], s, 3)

See also

Ways to use symmetricRing:

  • symmetricRing(Ring,ZZ)
  • symmetricRing(ZZ)
  • symmetricRing(Ring) -- The Symmetric ring corresponding to a given (Schur) ring.

For the programmer

The object symmetricRing 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.m2:5312:0.