Known Picard groups

Unless stated otherwise, ${\rm Pic}$ will always mean ${\rm Pic}_{\mathcal{O}}$.

Let $B$ be a block of $\mathcal{O}G$ for a finite group $G$. Let $D$ be a defect group and let $E$ be the inertial quotient.

Blocks with cyclic groups

Suppose that $D$ is cyclic.

If $E=1$, then $B$ is nilpotent and so ${\rm Pic}(B)\cong D:{\rm Aut}(D)$.

If $E \neq 1$, then ${\rm Pic}(B) \cong {\rm Out}_D(A) \times {\rm Aut}(D)/E$, where $A$ is a source algebra for $B$. Note that while ${\rm Out}_D(A)$ is isomorphic to a subgroup of $E$ and [BKL18, Theorem 1.4] gives a generator of this subgroup, it still needs to be computed on a case-by-case basis.

If follows from [BKL18, Theorem 1.4] that ${\rm Picent(B)}=1$.

Blocks with Klein four defect groups

Suppose that $D \cong C_2 \times C_2$.

Then $B$ is Morita equivalent to $\mathcal{O}D$, $\mathcal{O}A_4$ or $B_0(\mathcal{O}A_5)$. The Picard groups are known by [BKL18].

${\rm Pic}(\mathcal{O}D) = \mathcal{L}(B) \cong D:{\rm Aut(D)} \cong D:S_3$.

${\rm Pic}(\mathcal{O}A_4) = \mathcal{T}(B) \cong S_3$.

${\rm Pic}(B_0(\mathcal{O}A_5)) = \mathcal{T}(B) \cong C_2$.

Note that since any block with this defect group is source algebra equivalent to one of these three blocks, it follows that every Morita self-equivalence is given by a linear source bimodule since the same is true for the three blocks.

Unpublished elementary calculations give that

${\rm Pic}_k(kD) \cong (k \times k):GL_2(k)$.

${\rm Pic}_k(kA_4) \cong (k^* \times k^* \times C_3):C_2$.

${\rm Pic}_k(B_0(kA_5)) = (k^* \times k^*):C_2$.

Blocks with normal defect groups

In [Liv19] it is shown that if $D \triangleleft G$, then ${\rm Pic}(B) = \mathcal{L}(B)$ and ${\rm Picent}(B)=1$.

Abelian defect groups of $2$-rank at most three for $p=2$

In [EL18c] the isomorphism type of the Picard groups is calculated for every $2$-block with abelian defect groups of $2$-rank at most three, with the exception of the principal block of the sporadic group $J_1$. This final case has been calculated by Eisele. In every case ${\rm Picent}(B)=1$.

Miscallaneous other cases

${\rm Pic}(B_0(\mathcal{O}(A_5 \times A_4))) \cong C_2 \times S_3$

${\rm Pic}(B_0(\mathcal{O} SL_2(2^n))) \cong C_n$ when $n \geq 2$