SchurRingIndexedVariableTable

Description

It is what makes the partition-indexed notation s_{3,2,1} or s_(3,2,1) work: when you write s in a Schur ring, s is bound to a SchurRingIndexedVariableTable whose _ method accepts a list, a sequence, or an integer and returns the corresponding Schur-basis element. The same mechanism drives the partition-indexed notation in every flavor of SchurRing -- monomial, symplectic, orthogonal, symmetric-group, and rational-GL variants all share this indexing interface.

i1 : S = schurRing(QQ, s, 4);

i2 : class s

o2 = SchurRingIndexedVariableTable

o2 : Type

i3 : s_{2,1}

o3 = s
      2,1

o3 : S

i4 : s_(2,1)

o4 = s
      2,1

o4 : S

i5 : s_2

o5 = s
      2

o5 : S

Each Schur ring comes with its own indexed variable table, and partitions with many parts print in the same compact way.

i6 : T = schurRing(QQ, t, 5);

i7 : t_{4,3,2,1}

o7 = t
      4,3,2,1

o7 : T

i8 : t_{1,1,1,1}

o8 = t
      1,1,1,1

o8 : T

i9 : dim t_{4,3,2,1}

o9 = 1024

For rings with Basis => "Monomial", the same table selects the monomial-symmetric-function basis elements instead.

i10 : M = schurRing(QQ, m, 4, Basis => "Monomial");

i11 : m_{2,1}

o11 = m
       2,1

o11 : M

i12 : m_{2,1} * m_{1}

o12 = m    + 2m    + 2m
       3,1     2,2     2,1,1

o12 : M

In the rational-GL flavor, subscripts are pairs of lists encoding positive and negative weights.

i13 : R = schurRing(QQ, r, 3, GroupActing => "RatGL");

i14 : r_({2,1},{1})

o14 = r
       {{2, 1}, {1}}

o14 : R

See also

• IndexedVariableTable
• SchurRing -- The class of all Schur rings
• schurRing -- Make a SchurRing