# symmetricRing -- Make a Symmetric ring

## Synopsis

• Usage:
symmetricRing(A,n)
symmetricRing n
• Inputs:
• Optional inputs:
• EHPVariables => ..., default value (e,h,p), Specifies sequence of symbols representing e-, h-, and p-functions
• GroupActing => ..., default value GL, Specifies the group that is acting
• SVariable => ..., default value s, Specifies symbol representing s-functions

## 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