# Notation

[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]kG[/math] or [math]\mathcal{O}G[/math] depending on the context.

[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 |