# Difference between revisions of "Notation"

$(K,\mathcal{O},k)$ is a $p$-modular system, where $\mathcal{O}$ is a complete discrete valuation ring with algebraically closed residue field $k=\mathcal{O}/J(\mathcal{O})$ and $K$ is the field of fractions of $\mathcal{O}$, of characteristic zero. When we need to make a consistent choice of $(K,\mathcal{O},k)$ we take $k$ to be the algebraic closure of the field with $p$ elements and $\mathcal{O}$ to be the ring of Witt vectors for $k$ This has the disadvantage that for $G$ a finite group $KG$ need not contain the primitive character idempotents, but this condition can usually be avoided. In general however the choice of $\mathcal{O}$ is not consistent across the literature and some care has to be taken.
In the below, $G$ is a finite group and $B$ is a block of $\mathcal{O}G$. If it is clear from context, $B$ may also mean the corresponding block of $kG$. When it is not otherwise clear from context $kB$ will refer to the block of $kG$.
 $k(B)$ Number of irreducible characters in $B$, equal to $\dim_k(Z(kB))$ $k_i(B)$ Number of irreducible characters in $B$ of height $i$ $l(B)$ Number of isomorphism classes of simple $B$-modules ${\rm mf_k(B)}$ The Morita-Frobenius number of $kB$ [Ke04] ${\rm mf_\mathcal{O}(B)}$ The $\mathcal{O}$-Morita Frobenius number ${\rm Pic}_\mathcal{O}(B)$ The Picard group of $B$ ${\rm Pic}_k(B)$ The Picard group of $kB$ $\mathcal{T}(B)$ The subgroup of ${\rm Pic}_\mathcal{O}(B)$ consisting of trivial source bimodules [BKL18] $\mathcal{L}(B)$ The subgroup of ${\rm Pic}_\mathcal{O}(B)$ consisting of linear source bimodules [BKL18] $\mathcal{E}(B)$ The subgroup of ${\rm Pic}_\mathcal{O}(B)$ consisting of endopermutation source bimodules [BKL18] $M(x,y,z)$ A $k$-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.