Cell Modules¶
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class
sage.modules.with_basis.cell_module.
CellModule
(A, mu, **kwds)¶ Bases:
sage.combinat.free_module.CombinatorialFreeModule
A cell module.
Let \(R\) be a commutative ring. Let \(A\) be a cellular \(R\)-algebra with cell datum \((\Lambda, i, M, C)\). A cell module \(W(\lambda)\) is the \(R\)-module given by \(R\{C_s \mid s \in M(\lambda)\}\) with an action of \(a \in A\) given by \(a C_s = \sum_{u \in M(\lambda)} r_a(u, s) C_u\), where \(r_a(u, s)\) is the same as those given by the cellular condition:
\[\begin{split}a C^{\lambda}_{st} = \sum_{u \in M(\lambda)} r_a(u, s) C_{ut}^{\lambda} + \sum_{\substack{\mu < \lambda \\ x,y \in M(\mu)}} R C^{\mu}_{xy}.\end{split}\]INPUT:
A
– a cellular algebramu
– an element of the cellular poset ofA
See also
AUTHORS:
Travis Scrimshaw (2015-11-5): Initial version
REFERENCES:
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class
Element
¶ Bases:
sage.modules.with_basis.indexed_element.IndexedFreeModuleElement
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bilinear_form
(x, y)¶ Return the bilinear form on
x
andy
.The cell module \(W(\lambda)\) has a canonical bilinear form \(\Phi_{\lambda} : W(\lambda) \times W(\lambda) \to W(\lambda)\) given by
\[\begin{split}C_{ss}^{\lambda} C_{tt}^{\lambda} = \Phi_{\lambda}(C_s, C_t) C_{st}^{\lambda} + \sum_{\substack{\mu < \lambda \\ x,y \in M(\mu)}} R C^{\mu}_{xy}.\end{split}\]EXAMPLES:
sage: S = SymmetricGroupAlgebra(QQ, 3) sage: W = S.cell_module([2,1]) sage: elt = W.an_element(); elt 2*W[[1, 2], [3]] + 2*W[[1, 3], [2]] sage: W.bilinear_form(elt, elt) 8
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bilinear_form_matrix
(ordering=None)¶ Return the matrix corresponding to the bilinear form of
self
.INPUT:
ordering
– (optional) an ordering of the indices
EXAMPLES:
sage: S = SymmetricGroupAlgebra(QQ, 3) sage: W = S.cell_module([2,1]) sage: W.bilinear_form_matrix() [1 0] [0 1]
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cellular_algebra
()¶ Return the cellular algebra of
self
.EXAMPLES:
sage: S = SymmetricGroupAlgebra(QQ, 3) sage: W = S.cell_module([2,1]) sage: W.cellular_algebra() is S.cellular_basis() True sage: S.has_coerce_map_from(W.cellular_algebra()) True
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nonzero_bilinear_form
()¶ Return
True
if the bilinear form ofself
is non-zero.EXAMPLES:
sage: S = SymmetricGroupAlgebra(QQ, 3) sage: W = S.cell_module([2,1]) sage: W.nonzero_bilinear_form() True
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radical
()¶ Return the radical of
self
.Let \(W(\lambda)\) denote a cell module. The radical of \(W(\lambda)\) is defined as
\[\operatorname{rad}(\lambda) := \{x \in W(\lambda) \mid \Phi_{\lambda}(x, y)\},\]and note that it is a submodule of \(W(\lambda)\).
EXAMPLES:
sage: S = SymmetricGroupAlgebra(QQ, 3) sage: W = S.cell_module([2,1]) sage: R = W.radical(); R Radical of Cell module indexed by [2, 1] of Cellular basis of Symmetric group algebra of order 3 over Rational Field sage: R.basis() Finite family {}
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radical_basis
()¶ Return a basis of the radical of
self
.EXAMPLES:
sage: S = SymmetricGroupAlgebra(QQ, 3) sage: W = S.cell_module([2,1]) sage: W.radical_basis() ()
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simple_module
()¶ Return the corresponding simple module of
self
.Let \(W(\lambda)\) denote a cell module. The simple module \(L(\lambda)\) is defined as \(W(\lambda) / \operatorname{rad}(\lambda)\), where \(\operatorname{rad}(\lambda)\) is the radical of the bilinear form \(\Phi_{\lambda}\).
See also
EXAMPLES:
sage: S = SymmetricGroupAlgebra(QQ, 3) sage: W = S.cell_module([2,1]) sage: L = W.simple_module(); L Simple module indexed by [2, 1] of Cellular basis of Symmetric group algebra of order 3 over Rational Field sage: L.has_coerce_map_from(W) True
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class
sage.modules.with_basis.cell_module.
SimpleModule
(submodule)¶ Bases:
sage.modules.with_basis.subquotient.QuotientModuleWithBasis
A simple module of a cellular algebra.
Let \(W(\lambda)\) denote a cell module. The simple module \(L(\lambda)\) is defined as \(W(\lambda) / \operatorname{rad}(\lambda)\), where \(\operatorname{rad}(\lambda)\) is the radical of the bilinear form \(\Phi_{\lambda}\).
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class
Element
¶ Bases:
sage.modules.with_basis.indexed_element.IndexedFreeModuleElement
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class