Cell modules¶

class sage.modules.with_basis.cell_module.CellModule(A, mu, **kwds)[source]¶

Bases: 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[source]¶: Bases: IndexedFreeModuleElement

bilinear_form(x, y)[source]¶

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

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

Python

>>> from sage.all import *
>>> S = SymmetricGroupAlgebra(QQ, Integer(3))
>>> W = S.cell_module([Integer(2),Integer(1)])
>>> elt = W.an_element(); elt
2*W[[1, 2], [3]] + 2*W[[1, 3], [2]]
>>> W.bilinear_form(elt, elt)
8

bilinear_form_matrix(ordering=None)[source]¶

Return the matrix corresponding to the bilinear form of self.

INPUT:

ordering – (optional) an ordering of the indices

EXAMPLES:

Sage

sage: S = SymmetricGroupAlgebra(QQ, 3)
sage: W = S.cell_module([2,1])
sage: W.bilinear_form_matrix()
[1 0]
[0 1]

Python

>>> from sage.all import *
>>> S = SymmetricGroupAlgebra(QQ, Integer(3))
>>> W = S.cell_module([Integer(2),Integer(1)])
>>> W.bilinear_form_matrix()
[1 0]
[0 1]

cellular_algebra()[source]¶

Return the cellular algebra of self.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> S = SymmetricGroupAlgebra(QQ, Integer(3))
>>> W = S.cell_module([Integer(2),Integer(1)])
>>> W.cellular_algebra() is S.cellular_basis()
True
>>> S.has_coerce_map_from(W.cellular_algebra())
True

nonzero_bilinear_form()[source]¶

Return True if the bilinear form of self is nonzero.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> S = SymmetricGroupAlgebra(QQ, Integer(3))
>>> W = S.cell_module([Integer(2),Integer(1)])
>>> W.nonzero_bilinear_form()
True

radical()[source]¶

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

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 {}

Python

>>> from sage.all import *
>>> S = SymmetricGroupAlgebra(QQ, Integer(3))
>>> W = S.cell_module([Integer(2),Integer(1)])
>>> 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
>>> R.basis()
Finite family {}

radical_basis()[source]¶

Return a basis of the radical of self.

EXAMPLES:

Sage

sage: S = SymmetricGroupAlgebra(QQ, 3)
sage: W = S.cell_module([2,1])
sage: W.radical_basis()
()

Python

>>> from sage.all import *
>>> S = SymmetricGroupAlgebra(QQ, Integer(3))
>>> W = S.cell_module([Integer(2),Integer(1)])
>>> W.radical_basis()
()

simple_module()[source]¶

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 \(\Phi_{\lambda}\).