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r""" 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 """
#***************************************************************************** # Copyright (C) 2010 Eric Webster # Copyright (C) 2011 Hugh Thomas <hugh.ross.thomas@gmail.com> # # This program is free software: you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by # the Free Software Foundation, either version 2 of the License, or # (at your option) any later version. # http://www.gnu.org/licenses/ #*****************************************************************************
r""" Internal function called by :func:`schur_representation_indices`, which generates all weakly increasing tuples of length ``r`` in the alphabet ``1, 2, ..., n``.
EXAMPLES::
sage: from sage.algebras.schur_algebra import _schur_I_nr_representatives sage: _schur_I_nr_representatives(2, 4) ((1, 1, 1, 1), (1, 1, 1, 2), (1, 1, 2, 2), (1, 2, 2, 2), (2, 2, 2, 2)) """ return ()
continue
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))] """ else: else: else:
""" Simultaneously reorder a pair of tuples to obtain the equivalent element of the distinguished basis of the Schur algebra.
.. SEEALSO::
:func:`schur_representative_indices`
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)) """
r""" 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:
- [Gr2007]_ - :wikipedia:`Schur_algebra` """ """ Initialize ``self``.
TESTS::
sage: S = SchurAlgebra(ZZ, 2, 2) sage: TestSuite(S).run()
::
sage: SchurAlgebra(ZZ, -2, 2) Traceback (most recent call last): ... ValueError: n (=-2) must be a positive integer sage: SchurAlgebra(ZZ, 2, -2) Traceback (most recent call last): ... ValueError: r (=-2) must be a non-negative integer sage: SchurAlgebra('niet', 2, 2) Traceback (most recent call last): ... ValueError: R (=niet) must be a commutative ring """
schur_representative_indices(n, r), prefix='S', bracket=False, category=AlgebrasWithBasis(R).FiniteDimensional())
""" Return the string representation of ``self``.
EXAMPLES::
sage: SchurAlgebra(ZZ, 4, 4) Schur algebra (4, 4) over Integer Ring """
def one(self): """ 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 """ min_slope=0)
r""" 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)) """
# Find basis elements (p,q) such that p ~ i and q ~ l
# Find s in I(n,r) such that (p,s) ~ (i,j) and (s,q) ~ (k,l) and schur_representative_from_index(s, e[1]) == e_kl):
r""" Return the dimension of ``self``.
The dimension of the Schur algebra `S_R(n, r)` is
.. MATH::
\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 """
r""" 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
.. MATH::
(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 """ """ Initialize ``self``.
TESTS::
sage: T = SchurTensorModule(QQ, 2, 3) sage: TestSuite(T).run() """
""" Return the string representation of ``self``.
EXAMPLES::
sage: SchurTensorModule(QQ, 2, 3) The 3-fold tensor product of a free module of dimension 2 over Rational Field """
""" Result of acting by the basis element ``xi`` of the corresponding Schur algebra on the basis element ``v`` of ``self``.
EXAMPLES::
sage: T = SchurTensorModule(QQ, 2, 3) sage: xi = T._schur.basis().keys()[4]; xi ((1, 1, 2), (1, 1, 1)) sage: T._monomial_product(xi, (1, 1, 1)) B[1] # B[1] # B[2] + B[1] # B[2] # B[1] + B[2] # B[1] # B[1] """
""" Return the action of ``elt`` on ``self``.
We add the *left* action of the Schur algebra, and the *right* actions of the symmetric group algebra and the symmetric group.
EXAMPLES::
sage: T = SchurTensorModule(QQ, 2, 4) sage: x = T.an_element() sage: A = SchurAlgebra(QQ, 2, 4) sage: y = A.an_element() sage: y * x 14*B[1] # B[1] # B[1] # B[1] sage: x * y Traceback (most recent call last): ... TypeError: unsupported operand parent(s) for *: ...
::
sage: SGA = SymmetricGroupAlgebra(QQ, 4) sage: y = SGA.an_element() sage: x * y 14*B[1] # B[1] # B[1] # B[1] + 17*B[1] # B[1] # B[1] # B[2] + 7*B[1] # B[1] # B[2] # B[1] + 9*B[1] # B[2] # B[1] # B[1] + 2*B[2] # B[1] # B[1] # B[1] sage: y * x Traceback (most recent call last): ... TypeError: unsupported operand parent(s) for *: ...
::
sage: S = Permutations(4) sage: y = S.an_element() sage: x * y 2*B[1] # B[1] # B[1] # B[1] + 3*B[1] # B[1] # B[1] # B[2] + 2*B[2] # B[1] # B[1] # B[1] """ c * ce) for m, c in self for me, ce in elt)
for m, c in self)
r""" 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] """
#make ST the superstandard tableau of shape mu
#make ell the reading word of the highest weight tableau of shape mu
# This is the notation `\{X\}` from just before (5.3a) of [Gr2007]_. remove_zeros=False)
# [Green, Theorem 5.3b] says that a basis of the Carter-Lusztig # module V_\mu is given by taking this f, and multiplying by all # xi_{i,ell} with ell as above and i semistandard.
#Therefore, we now have carter_lusztig as a list giving the basis #of `V_\mu`
#We want to think of expressing this character as a sum of monomial #symmetric functions.
#We will determine a basis element for each m_\lambda in the #character, and we want to keep track of them by \lambda.
#That means that we only want to pick out the basis elements above for #those semistandard words whose content is a partition.
# all partitions of r, length at most n
# JJ will consist of a list for each element of `contents`, # recording the list # of semistandard tableaux words with that content
# graded_basis will consist of the a corresponding basis element # Get the content of T
#There is an inner product on the Carter-Lusztig module V_\mu; its #maximal submodule is exactly the kernel of the inner product.
#Now, for each possible partition content, we look at the graded piece of #that degree, and we record how these elements pair with each of the #elements of carter_lusztig.
#The kernel of this pairing is the part of this graded piece which is #not in the irreducible module for \mu.
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