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 2 2 o2 = s + (2x + 1)s + (x + 2x + 1)s + (x + 2x)s + (x + 1)s + (2x + 2x)s + x s 6 5,1 4,2 4,1,1 3,3 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 |
The object schurRing is a method function with options.