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: 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: 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: IndexedFreeModuleElement

construction()#

Return None.

There is no functorial construction for self.

EXAMPLES:

sage: T = SchurTensorModule(QQ, 2, 3)
sage: T.construction()

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.