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# schurRing -- Make a SchurRing

## Synopsis

• 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

S = schurRing(A,s,n) creates a Schur ring of degree n over the base ring A, with variables based on the symbol s. This is the representation ring for the general linear group of n by n matrices, tensored with the ring A. If s is already assigned a value as a variable in a ring, its base symbol will be used, if it is possible to determine.

 i1 : S = schurRing(QQ[x],s,3); i2 : (x*s_{2,1}+s_3)^2 2 2 2 o2 = s + (2x + 1)s + (x + 2x + 1)s + (x + 2x)s + (x + 1)s + 6 5,1 4,2 4,1,1 3,3 ------------------------------------------------------------------------ 2 2 (2x + 2x)s + x s 3,2,1 2,2,2 o2 : S

Alternatively, the elements of a Schur ring may be interpreted as characters of symmetric groups. To indicate this interpretation, one has to set the value of the option GroupActing to "Sn".

 i3 : S = schurRing(s,4,GroupActing => "Sn"); i4 : exteriorPower(2,s_(3,1)) o4 = s 2,1,1 o4 : S

If the dimension n is not specified, then one should think of S as the full ring of symmetric functions over the base A, i.e. there is no restriction on the number of parts of the partitions indexing the generators of S.

 i5 : S = schurRing(ZZ/5,t) o5 = S o5 : SchurRing i6 : (t_(2,1)-t_3)^2 o6 = t - t - t + 2t + t + t + t 6 5,1 4,1,1 3,3 3,1,1,1 2,2,2 2,2,1,1 o6 : S

If the base ring A is not specified, then QQ is used instead.

 i7 : S = schurRing(r,2,EHPVariables => (re,rh,rp)) o7 = S o7 : SchurRing i8 : toH r_(2,1) 3 o8 = rh - rh rh 1 1 2 o8 : QQ[re ..re , rp ..rp , rh ..rh ] 1 2 1 2 1 2

## Ways to use schurRing :

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

## For the programmer

The object schurRing is .