Difference between revisions of "Known Picard groups"
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Unless stated otherwise, <math>{\rm Pic}</math> will always mean <math>{\rm Pic}_{\mathcal{O}}</math>. | Unless stated otherwise, <math>{\rm Pic}</math> will always mean <math>{\rm Pic}_{\mathcal{O}}</math>. | ||
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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=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>.<ref>See Theorem 1.4 of [[References #B|[BKL18]]]</ref> Note that <math>{\rm Out}_D(A)</math> is isomorphic to a subgroup of <math>E</math> and [[References#B|[BKL18, Theorem 1.4]]] gives a generator of this subgroup, it still needs to be computed on a case-by-case basis. | + | 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>.<ref>See Theorem 1.4 of [[References #B|[BKL18]]]</ref> Note that while <math>{\rm Out}_D(A)</math> is isomorphic to a subgroup of <math>E</math> and [[References#B|[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 [[References#B|[BKL18, Theorem 1.4]]] that <math>{\rm Picent(B)}=1</math>. | If follows from [[References#B|[BKL18, Theorem 1.4]]] that <math>{\rm Picent(B)}=1</math>. | ||
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Unpublished elementary calculations give that | Unpublished elementary calculations give that | ||
| − | <math>{\rm Pic}(kD) \cong (k \times k):GL_2(k)</math>. | + | <math>{\rm Pic}_k(kD) \cong (k \times k):GL_2(k)</math>. |
| − | <math>{\rm Pic}( | + | <math>{\rm Pic}_k(kA_4) \cong (k^* \times k^* \times C_3):C_2</math>. |
| − | <math>{\rm Pic}(B_0( | + | <math>{\rm Pic}_k(B_0(kA_5)) = (k^* \times k^*):C_2</math>. |
== Blocks with normal defect groups == | == Blocks with normal defect groups == | ||
| − | In [[References#L|[Liv19]]] it is shown that if <math>D \triangleleft G</math>, then <math>{\rm Pic}(B) = \mathcal{L}(B)</math> and <math>{\rm Picent}(B)=1</math>. | + | In [[References#L|[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 [[References#L|[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> == | == Abelian defect groups of <math>2</math>-rank at most three for <math>p=2</math> == | ||
In [[References#E|[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>. | In [[References#E|[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>. | ||
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| + | == Miscallaneous other cases == | ||
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| + | <math>{\rm Pic}(B_0(\mathcal{O}(A_5 \times A_4))) \cong C_2 \times S_3</math><ref>See [[References#E|[EL18c]]]</ref> | ||
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| + | <math>{\rm Pic}(B_0(\mathcal{O} SL_2(2^n))) \cong C_n</math> when <math>n \geq 2</math><ref>See [[References#E|[EL18c]]]</ref> | ||
== Notes == | == Notes == | ||
<references /> | <references /> | ||
Latest revision as of 11:04, 25 May 2021
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.
Contents
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].[1] 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[2], 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][3]
[math]{\rm Pic}(B_0(\mathcal{O} SL_2(2^n))) \cong C_n[/math] when [math]n \geq 2[/math][4]
