Known Picard groups

Unless stated otherwise, [math]{\rm Pic}[/math] will always mean [math]{\rm Pic}_{\mathcal{O}}[/math].

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

Blocks with cyclic groups

Suppose that [math]D[/math] is cyclic.

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

If [math]E \neq 1[/math], then [math]{\rm Pic}(B) \cong {\rm Out}_D(A) \times {\rm Aut}(D)/E[/math], where [math]A[/math] is a source algebra for [math]B[/math]. Note that while [math]{\rm Out}_D(A)[/math] is isomorphic to a subgroup of [math]E[/math] 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 [math]{\rm Picent(B)}=1[/math].

Blocks with Klein four defect groups

Suppose that [math]D \cong C_2 \times C_2[/math].

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

[math]{\rm Pic}(\mathcal{O}D) = \mathcal{L}(B) \cong D:{\rm Aut(D)} \cong D:S_3[/math].

[math]{\rm Pic}(\mathcal{O}A_4) = \mathcal{T}(B) \cong S_3[/math].

[math]{\rm Pic}(B_0(\mathcal{O}A_5)) = \mathcal{T}(B) \cong C_2[/math].

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Unpublished elementary calculations give that

[math]{\rm Pic}_k(kD) \cong (k \times k):GL_2(k)[/math].

[math]{\rm Pic}_k(kA_4) \cong (k^* \times k^* \times C_3):C_2[/math].

[math]{\rm Pic}_k(B_0(kA_5)) = (k^* \times k^*):C_2[/math].

Blocks with normal defect groups

In [Liv19] it is shown that if [math]D \triangleleft G[/math] and [math]E[/math] is abelian, then [math]{\rm Pic}(B) = \mathcal{L}(B)[/math] and [math]{\rm Picent}(B)=1[/math]. In [LiMa20b], it was shown that for odd primes [math]{\rm Pic}(B) = \mathcal{L}(B)[/math] whenever [math]D \triangleleft G[/math], and that this result holds for [math]p=2[/math] if we further assume that [math]D/[D,D][/math] has no direct factor isomorphic to [math]C_2[/math].

Abelian defect groups of [math]2[/math]-rank at most three for [math]p=2[/math]

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

Miscallaneous other cases

[math]{\rm Pic}(B_0(\mathcal{O}(A_5 \times A_4))) \cong C_2 \times S_3[/math]

[math]{\rm Pic}(B_0(\mathcal{O} SL_2(2^n))) \cong C_n[/math] when [math]n \geq 2[/math]