On Brauer trees of blocks of finite groups of Lie type Radha Kessar - PowerPoint PPT Presentation

On Brauer trees of blocks of finite groups of Lie type Radha Kessar City, University of London joint work with David Craven University of Birmingham June 13, 2019

Motivation: Describe the Brauer tree algebras which occur as blocks of finite group algebras. Upgrade the Jordan decomposition in the character theory of finite groups of Lie type to categorical equivalences.

Brauer tree: finite tree along with prescribed cyclic ordering of the edges around each vertex. assignment of a positive integer called the multiplicity to one of the vertices, called the exceptional vertex . ◦ ◦ ◦ • ◦ ◦ ◦ ◦ anti-clockwise ordering, multiplicity =2

T : Brauer tree, k : field Brauer tree algebra: Finite dimensional k -algebra A , Edges of T labelled by the isomorphism classes of simple A modules. Projective cover of simple A -module S is read off from the edges covalent to S . If exceptional multiplicity is m , then for edges incident to the exceptional vertex need to “go around” m -times.

◦ ◦ 1 2 ◦ • ◦ ◦ ◦ 3 4 5 6 7 ◦ anti-clockwise ordering, multiplicity =2 Projective cover of 4: 4 1 3 4 ⊕ 5 1 3 4

Blocks G : finite group ℓ : prime number O : complete discrete valuation ring of characteristic 0 and residue field k of characteristic ℓ , assumed large enough B : block of O G , ¯ B := k ⊗ O B corresponding block of kG P : defect group of B Irr ( B ): subset of ordinary irreducible characters of G belonging to B Basic question of block theory : Describe structure of B and ¯ B in terms of P . P cyclic if and only if ¯ B is of finite representation type. In this case, ¯ B is a Brauer tree algebra. If P cyclic, then the Morita equivalence class of ¯ B determines the Morita equivalence class of B .

Suppose that the block B with cyclic defect P has tree T with e edges and multiplicity m . e | ℓ − 1 and em = | P | − 1 . The vertices of T are labelled by elements of Irr ( B ); vertex edge incidence determines decomposition numbers; exceptional vertex corresponds to an m -pack of irreducible characters; | Irr ( B ) | = e + m . ◦ ◦ ◦ • ◦ ◦ ◦ ◦ anti-clockwise ordering, multiplicity =2 Not a block as 7 × 2 = 15 − 1 and 15 is not a prime power.

Question: Which Brauer trees occur as blocks of finite group algebras? An n - unfolding of a Brauer tree is n -copies of the tree identified at one vertex. Two Brauer trees are similar if both are unfoldings of a common Brauer tree. Every cyclic block is similar to a cyclic block of a finite quasi-simple group (Feit ’84) So, up to unfolding, it suffices to know what happens for finite quasisimple groups. “ As far as I know not a single Brauer tree can be eliminated without using the classification of finite simple groups ”(Feit ’84)

Some guiding principles. Folding. Let G ✂ L , C block of O L such that there exists χ ∈ Irr ( C ), ψ ∈ Irr ( B ) with � Ind L G ψ, χ � � = 0. Suppose that | L / G | =: n � = ℓ is prime. If B and C have the same number of simple modules, then B and C are Morita equivalent, hence have the same tree. If C has more simple modules than B , then the tree of C is an n -unfolding of the tree of B . If B has more simple modules than C , then the tree of B is an n -unfolding of the tree of C . So, may replace G with more convenient versions, e.g. alternating groups with symmetric groups.

Good bijections. Let L be a finite group and C a cyclic block of O L with defect group P . Suppose that there exists an ( O G , O L )-bimodule F finitely generated and projective as left and right module a bijection χ → ˜ χ between Irr ( B ) and Irr ( C ) respecting exceptionality a positive integer r such that F ( χ ) = r ˜ χ, ∀ χ ∈ Irr ( B ) . Then B and C are Morita equivalent, hence have the same tree. Cyclic blocks of quasi-simple groups are replete with good bijections, e.g. any two cyclic blocks of symmetric groups with the same defect group can be linked to each other by a sequence of good bijections (with F a summand of induction/restriction), hence have the same tree.

Reality. Suppose that all χ ∈ Irr ( B ) are real valued. Then the Brauer tree of B is a straight line. Quasisimple groups have strong reality properties ❀ The Brauer tree of any finite classical group, and of any symmetric and alternating group is similar to a line. (Feit ’84) Character degree ratios. If χ, ψ adjacent non-exceptional vertices, then χ (1) ψ (1) ≡ − 1 mod ℓ.

Brauer trees of finite quasisimple groups. Alternating groups: (M¨ uller (2001)). Sporadic groups: Shapes are known except for the Baby Monster and the Monster (Thackeray (1981), Humphreys (1982), Benson (1985), Feit (1985), Hiss-Lux (1988), Wilson (1993), Cooperman-Hiss-Lux-M¨ uller (1997), Ottenmann (2000), R¨ ohr (2000), M¨ ueller-Neunh¨ offer-R¨ ohr-Wilson (2002), Naehrig (2002). In addition, in a few cases there are unresolved issues regarding cyclic ordering around the vertices and labelling of characters.

Groups of Lie type Let G = G ( q ), finite quasisimple group of Lie type. ℓ | q : the only possibility is G = PSL 2 ( p ), B is the principal block, Brauer tree is easy to determine (straight line). From now on, assume ℓ ∤ q . G of classical type: Known modulo a few exceptions (Fong and Srinivasan, (1984), (1990)). G one of G 2 ( q ), 2 G 2 ( q ), 2 F 4 ( q 2 ), 3 D 4 ( q ), 2 B 2 ( q ): (Hiss, Lux, Geck, Shamash, White, Wings (1990s)). G = E 6 ( q ): (Hiss-L´ ’ubeck-Malle (1995)). Left with F 4 ( q ), 2 E 6 ( q ), E 7 ( q ), E 8 ( q ).

Unipotent blocks G = G ( q ) = G F , G connected reductive group over ¯ F q , F : G → G Frobenius morphism. An irreducible character χ of G is unipotent if � R G T (1) , χ � � = 0 for some F -stable maximal torus T of G , where 1 is the trivial character of T F and R G T denotes Deligne-Lusztig induction. A block which contains a unipotent character is a unipotent block . Theorem (Hiss-L¨ ubeck (1998), Dudas (2012), Dudas-Rouquier (2014), Craven-Dudas-Rouquier (JEMS, to appear)). Suppose that G is one of F 4 ( q ), 2 E 6 ( q ), E 7 ( q ), E 8 ( q ) and B is a unipotent cyclic block of O G .Then the Brauer tree of B is known. Unipotent blocks OK.

From unipotent blocks to general blocks: Jordan decomposition G ∗ = G ∗ F , dual group. G = G ( q ) = G F , The conjugacy classes of semi-simple elements of G ∗ partition Irr ( G ): � Irr ( G ) = E ( G , s ) (Lusztig series) s E ( G , 1) is the subset of unipotent characters. Assume Z ( G ) is connected. Then E ( G , s ) ↔ E ( C G ∗ ( s ) , 1) (Jordan correspondence)

Theorem (Enguehard (2009)). Suppose ℓ ≥ 7, G has connected center, [ G , G ] is simple, simply connected and [ G , G ] F not one of 2 E 6 (2), E 7 (2), E 8 (2). Let B be a block of O G . There exists a semi-simple element s ∈ G ∗ , a finite group G ( s ) of Lie type in duality with C G ∗ ( s ) and a unipotent block B ( s ) of G ( s ) such that Jordan correspondence induces a bijection between Irr ( B ) and Irr ( B ( s )). B and B ( s ) have isomorphic defect groups. Theorem (Craven-K. (2019)). With the notation and assumptions of Enguehard’s theorem, assume B is cyclic. Assume further that if G = E 8 ( q ), then C G ∗ ( s ) is not of type 2 E 6 ( q ) 2 A 2 ( q ). Then B and B ( s ) have the same Brauer trees.