Known Picard groups - Revision history

Charles Eaton: /* Blocks with normal defect groups */

2021-05-25T11:04:14Z

‎Blocks with normal defect groups

CesareGArdito: /* Blocks with normal defect groups */

2019-12-08T18:53:29Z

‎Blocks with normal defect groups

Charles Eaton at 10:19, 13 August 2019

2019-08-13T10:19:36Z

Charles Eaton at 16:27, 12 August 2019

2019-08-12T16:27:54Z

Charles Eaton at 14:43, 12 August 2019

2019-08-12T14:43:04Z

Charles Eaton: Created page with "This page will detail cases where the Picard groups are known."

2018-11-23T14:21:33Z

Created page with "This page will detail cases where the Picard groups are known."

New page

This page will detail cases where the Picard groups are known.

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 == Blocks with normal defect groups ==
-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|[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> ==

@@ Line 37: / Line 37: @@
 == 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>.
 == Abelian defect groups of <math>2</math>-rank at most three for <math>p=2</math> ==

@@ Line 9: / Line 9: @@
 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>.
@@ Line 29: / Line 29: @@
 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}(\mathcal{O}A_4) \cong (k^* \times k^* \times C_3):C_2</math>.
+<math>{\rm Pic}_k(kA_4) \cong (k^* \times k^* \times C_3):C_2</math>.
-<math>{\rm Pic}(B_0(\mathcal{O}A_5)) = (k^* \times k^*):C_2</math>.
+<math>{\rm Pic}_k(B_0(kA_5)) = (k^* \times k^*):C_2</math>.
 == Blocks with normal defect groups ==

@@ Line 1: / Line 1: @@
 Unless stated otherwise, <math>{\rm Pic}</math> will always mean <math>{\rm Pic}_{\mathcal{O}}</math>.
@@ Line 44: / Line 42: @@
 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>.
 == Notes ==
 <references />

← Older revision		Revision as of 14:43, 12 August 2019
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−	~~This page~~ will ~~detail cases~~ where ~~the~~ Picard groups are known.	+	[[Image:under-construction.png\|50px\|left]]
		+
		+	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>.<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 follows from [[References#B\|[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 [[References#B\|[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<ref>See [[References#C\|[CEKL11]]]</ref>, 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}(kD) \cong (k \times k):GL_2(k)</math>.
		+
		+	<math>{\rm Pic}(\mathcal{O}A_4) \cong (k^* \times k^* \times C_3):C_2</math>.
		+
		+	<math>{\rm Pic}(B_0(\mathcal{O}A_5)) = (k^* \times k^*):C_2</math>.
		+
		+	== 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>.
		+
		+	== 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>.
		+
		+	== Notes ==
		+
		+	<references />