Schur algebras for \(GL_n\)

This file implements:

  • Schur algebras for \(GL_n\) over an arbitrary field.
  • The canonical action of the Schur algebra on a tensor power of the standard representation.
  • Using the above to calculate the characters of irreducible \(GL_n\) modules.

AUTHORS:

  • Eric Webster (2010-07-01): implement Schur algebra
  • Hugh Thomas (2011-05-08): implement action of Schur algebra and characters of irreducible modules
sage.algebras.schur_algebra.GL_irreducible_character(n, mu, KK)

Return the character of the irreducible module indexed by mu of \(GL(n)\) over the field KK.

INPUT:

  • n – a positive integer
  • mu – a partition of at most n parts
  • KK – a field

OUTPUT:

a symmetric function which should be interpreted in n variables to be meaningful as a character

EXAMPLES:

Over \(\QQ\), the irreducible character for \(\mu\) is the Schur function associated to \(\mu\), plus garbage terms (Schur functions associated to partitions with more than \(n\) parts):

sage: from sage.algebras.schur_algebra import GL_irreducible_character
sage: sbasis = SymmetricFunctions(QQ).s()
sage: z = GL_irreducible_character(2, [2], QQ)
sage: sbasis(z)
s[2]

sage: z = GL_irreducible_character(4, [3, 2], QQ)
sage: sbasis(z)
-5*s[1, 1, 1, 1, 1] + s[3, 2]

Over a Galois field, the irreducible character for \(\mu\) will in general be smaller.

In characteristic \(p\), for a one-part partition \((r)\), where \(r = a_0 + p a_1 + p^2 a_2 + \dots\), the result is (see [Gr2007], after 5.5d) the product of \(h[a_0], h[a_1]( pbasis[p]), h[a_2] ( pbasis[p^2]), \dots,\) which is consistent with the following

sage: from sage.algebras.schur_algebra import GL_irreducible_character
sage: GL_irreducible_character(2, [7], GF(3))
m[4, 3] + m[6, 1] + m[7]
class sage.algebras.schur_algebra.SchurAlgebra(R, n, r)

Bases: sage.combinat.free_module.CombinatorialFreeModule

A Schur algebra.

Let \(R\) be a commutative ring, \(n\) be a positive integer, and \(r\) be a non-negative integer. Define \(A_R(n,r)\) to be the set of homogeneous polynomials of degree \(r\) in \(n^2\) variables \(x_{ij}\). Therefore we can write \(R[x_{ij}] = \bigoplus_{r \geq 0} A_R(n,r)\), and \(R[x_{ij}]\) is known to be a bialgebra with coproduct given by \(\Delta(x_{ij}) = \sum_l x_{il} \otimes x_{lj}\) and counit \(\varepsilon(x_{ij}) = \delta_{ij}\). Therefore \(A_R(n,r)\) is a subcoalgebra of \(R[x_{ij}]\). The Schur algebra \(S_R(n,r)\) is the linear dual to \(A_R(n,r)\), that is \(S_R(n,r) := \hom(A_R(n,r), R)\), and \(S_R(n,r)\) obtains its algebra structure naturally by dualizing the comultiplication of \(A_R(n,r)\).

Let \(V = R^n\). One of the most important properties of the Schur algebra \(S_R(n, r)\) is that it is isomorphic to the endomorphisms of \(V^{\otimes r}\) which commute with the natural action of \(S_r\).

EXAMPLES:

sage: S = SchurAlgebra(ZZ, 2, 2); S
Schur algebra (2, 2) over Integer Ring

REFERENCES:

dimension()

Return the dimension of self.

The dimension of the Schur algebra \(S_R(n, r)\) is

\[\dim S_R(n,r) = \binom{n^2+r-1}{r}.\]

EXAMPLES:

sage: S = SchurAlgebra(QQ, 4, 2)
sage: S.dimension()
136
sage: S = SchurAlgebra(QQ, 2, 4)
sage: S.dimension()
35
one()

Return the element \(1\) of self.

EXAMPLES:

sage: S = SchurAlgebra(ZZ, 2, 2)
sage: e = S.one(); e
S((1, 1), (1, 1)) + S((1, 2), (1, 2)) + S((2, 2), (2, 2))

sage: x = S.an_element()
sage: x * e == x
True
sage: all(e * x == x for x in S.basis())
True

sage: S = SchurAlgebra(ZZ, 4, 4)
sage: e = S.one()
sage: x = S.an_element()
sage: x * e == x
True
product_on_basis(e_ij, e_kl)

Return the product of basis elements.

EXAMPLES:

sage: S = SchurAlgebra(QQ, 2, 3)
sage: B = S.basis()

If we multiply two basis elements \(x\) and \(y\), such that \(x[1]\) and \(y[0]\) are not permutations of each other, the result is zero:

sage: S.product_on_basis(((1, 1, 1), (1, 1, 2)), ((1, 2, 2), (1, 1, 2)))
0

If we multiply a basis element \(x\) by a basis element which consists of the same tuple repeated twice (on either side), the result is either zero (if the previous case applies) or \(x\):

sage: ww = B[((1, 2, 2), (1, 2, 2))]
sage: x = B[((1, 2, 2), (1, 1, 2))]
sage: ww * x
S((1, 2, 2), (1, 1, 2))

An arbitrary product, on the other hand, may have multiplicities:

sage: x = B[((1, 1, 1), (1, 1, 2))]
sage: y = B[((1, 1, 2), (1, 2, 2))]
sage: x * y
2*S((1, 1, 1), (1, 2, 2))
class sage.algebras.schur_algebra.SchurTensorModule(R, n, r)

Bases: sage.combinat.free_module.CombinatorialFreeModule_Tensor

The space \(V^{\otimes r}\) where \(V = R^n\) equipped with a left action of the Schur algebra \(S_R(n,r)\) and a right action of the symmetric group \(S_r\).

Let \(R\) be a commutative ring and \(V = R^n\). We consider the module \(V^{\otimes r}\) equipped with a natural right action of the symmetric group \(S_r\) given by

\[(v_1 \otimes v_2 \otimes \cdots \otimes v_n) \sigma = v_{\sigma(1)} \otimes v_{\sigma(2)} \otimes \cdots \otimes v_{\sigma(n)}.\]

The Schur algebra \(S_R(n,r)\) is naturally isomorphic to the endomorphisms of \(V^{\otimes r}\) which commutes with the \(S_r\) action. We get the natural left action of \(S_R(n,r)\) by this isomorphism.

EXAMPLES:

sage: T = SchurTensorModule(QQ, 2, 3); T
The 3-fold tensor product of a free module of dimension 2
 over Rational Field
sage: A = SchurAlgebra(QQ, 2, 3)
sage: P = Permutations(3)
sage: t = T.an_element(); t
2*B[1] # B[1] # B[1] + 2*B[1] # B[1] # B[2] + 3*B[1] # B[2] # B[1]
sage: a = A.an_element(); a
2*S((1, 1, 1), (1, 1, 1)) + 2*S((1, 1, 1), (1, 1, 2))
 + 3*S((1, 1, 1), (1, 2, 2))
sage: p = P.an_element(); p
[3, 1, 2]
sage: y = a * t; y
14*B[1] # B[1] # B[1]
sage: y * p
14*B[1] # B[1] # B[1]
sage: z = t * p; z
2*B[1] # B[1] # B[1] + 3*B[1] # B[1] # B[2] + 2*B[2] # B[1] # B[1]
sage: a * z
14*B[1] # B[1] # B[1]

We check the commuting action property:

sage: all( (bA * bT) * p == bA * (bT * p)
....:      for bT in T.basis() for bA in A.basis() for p in P)
True
class Element

Bases: sage.modules.with_basis.indexed_element.IndexedFreeModuleElement

sage.algebras.schur_algebra.schur_representative_from_index(i0, i1)

Simultaneously reorder a pair of tuples to obtain the equivalent element of the distinguished basis of the Schur algebra.

INPUT:

  • A pair of tuples of length \(r\) with elements in \(\{1,\dots,n\}\)

OUTPUT:

  • The corresponding pair of tuples ordered correctly.

EXAMPLES:

sage: from sage.algebras.schur_algebra import schur_representative_from_index
sage: schur_representative_from_index([2,1,2,2], [1,3,0,0])
((1, 2, 2, 2), (3, 0, 0, 1))
sage.algebras.schur_algebra.schur_representative_indices(n, r)

Return a set which functions as a basis of \(S_K(n,r)\).

More specifically, the basis for \(S_K(n,r)\) consists of equivalence classes of pairs of tuples of length r on the alphabet \(\{1, \dots, n\}\), where the equivalence relation is simultaneous permutation of the two tuples. We can therefore fix a representative for each equivalence class in which the entries of the first tuple weakly increase, and the entries of the second tuple whose corresponding values in the first tuple are equal, also weakly increase.

EXAMPLES:

sage: from sage.algebras.schur_algebra import schur_representative_indices
sage: schur_representative_indices(2, 2)
[((1, 1), (1, 1)), ((1, 1), (1, 2)),
 ((1, 1), (2, 2)), ((1, 2), (1, 1)),
 ((1, 2), (1, 2)), ((1, 2), (2, 1)),
 ((1, 2), (2, 2)), ((2, 2), (1, 1)),
 ((2, 2), (1, 2)), ((2, 2), (2, 2))]