Cell Modules

sage.modules.with_basis.cell_module.CellModule

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:

sage.modules.with_basis.cell_module.SimpleModule

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 \(\Phi_{\lambda}\).