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schurRing -- Make a SchurRing

Synopsis

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

See also

Ways to use schurRing :

For the programmer

The object schurRing is a method function with options.