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SchurRings :: symmetricRing

symmetricRing -- Make a Symmetric ring

Synopsis

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

See also

Ways to use symmetricRing :

For the programmer

The object symmetricRing is a method function with options.