Cell Modules

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 algebra
  • mu – an element of the cellular poset of A

See also

CellularBasis

AUTHORS:

  • Travis Scrimshaw (2015-11-5): Initial version

REFERENCES:

class Element

Bases: sage.modules.with_basis.indexed_element.IndexedFreeModuleElement

bilinear_form(x, y)

Return the bilinear form on x and y.

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
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]
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
nonzero_bilinear_form()

Return True if the bilinear form of self is non-zero.

EXAMPLES:

sage: S = SymmetricGroupAlgebra(QQ, 3)
sage: W = S.cell_module([2,1])
sage: W.nonzero_bilinear_form()
True
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 {}
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()
()
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

radical()

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
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 \(\Phi_{\lambda}\).

class Element

Bases: sage.modules.with_basis.indexed_element.IndexedFreeModuleElement