Notation

From Block library
Jump to: navigation, search

[math](K,\mathcal{O},k)[/math] is a [math]p[/math]-modular system, where [math]\mathcal{O}[/math] is a complete discrete valuation ring with algebraically closed residue field [math]k=\mathcal{O}/J(\mathcal{O})[/math] and [math]K[/math] is the field of fractions of [math]\mathcal{O}[/math], of characteristic zero. In order to make a consistent choice of [math](K,\mathcal{O},k)[/math] we take [math]k[/math] to be the algebraic closure of the field with [math]p[/math] elements and [math]\mathcal{O}[/math] to be the ring of Witt vectors for [math]k[/math] This has the disadvantage that for [math]G[/math] a finite group [math]KG[/math] need not contain the primitive character idempotents, but this condition can usually be avoided.

In the below, [math]G[/math] is a finite group and [math]B[/math] is a block of [math]\mathcal{O}G[/math]. If it is clear from context, [math]B[/math] may also mean the corresponding block of [math]kG[/math]. When it is not otherwise clear from context [math]kB[/math] will refer to the block of [math]kG[/math].

[math]k(B)[/math] Number of irreducible characters in [math]B[/math]
[math]l(B)[/math] Number of isomorphism classes of simple [math]B[/math]-modules
[math]{\rm mf_k(B)}[/math] The Morita-Frobenius number of [math]kB[/math] [Ke04]
[math]{\rm mf_\mathcal{O}(B)}[/math] The [math]\mathcal{O}[/math]-Morita Frobenius number
[math]{\rm Pic}_\mathcal{O}(B)[/math] The Picard group of [math]B[/math]
[math]{\rm Pic}_k(B)[/math] The Picard group of [math]kB[/math]
[math]\mathcal{T}(B)[/math] The subgroup of [math]{\rm Pic}_\mathcal{O}(B)[/math] consisting of trivial source bimodules [BKL18]
[math]\mathcal{L}(B)[/math] The subgroup of [math]{\rm Pic}_\mathcal{O}(B)[/math] consisting of linear source bimodules [BKL18]
[math]\mathcal{E}(B)[/math] The subgroup of [math]{\rm Pic}_\mathcal{O}(B)[/math] consisting of endopermutation source bimodules [BKL18]
[math]M(x,y,z)[/math] A [math]k[/math]-Morita equivalence class consisting of blocks with defect groups of order x, with a representative having defect group SmallGroup(x,y) in GAP/MAGMA labelling. It is the z-th such class.