aa r X i v : . [ m a t h . R T ] M a y Blocks with small-dimensional basic algebra
Benjamin Sambale ∗ May 28, 2020
Abstract
Linckelmann and Murphy have classified the Morita equivalence classes of p -blocks of finite groupswhose basic algebra has dimension at most . We extend their classification to dimension and . As predicted by Donovan’s Conjecture, we obtain only finitely many such Morita equivalenceclasses. Keywords: basic algebra of block, Morita equivalence, Donovan’s conjecture
AMS classification:
Let F be an algebraically closed field of characteristic p > . Donovan’s Conjecture (over F ) statesthat for every finite p -group D there are only finitely many Morita equivalence classes of p -blocks offinite groups with defect group D . Since a general proof seems illusive at present, mathematicians havefocused on certain families of p -groups D . This has culminated in a proof of Donovan’s Conjecturefor all abelian -groups by Eaton–Livesey [4]. A different approach, introduced by Linckelmann [11],aims to classify blocks B with a given basic algebra A . Recall that A is the unique F -algebra (up toisomorphism) of smallest dimension which is Morita equivalent to B . Linckelmann and Murphy [11, 12]have classified all blocks B such that dim A ≤ . Since the order of a defect group is bounded in termsof dim A (see next section), one expects only finitely many such blocks up to Morita equivalence. Indeedthe list in [11] is finite. We extend their classification as follows. Theorem 1.
Let B be a block of a finite group with basic algebra A .(I) If dim A = 13 , then B is Morita equivalent to one of the following block algebras:(a) F C ( p = 13 ).(b) the principal -block of PSL(3 , with defect .(c) the principal -block of PSL(2 , with defect .(d) the principal -block of PGL(2 , with defect group D .(e) a non-principal -block of .M with defect group SD .(f ) a non-principal -block of .A with defect . ∗ Institut für Algebra, Zahlentheorie und Diskrete Mathematik, Leibniz Universität Hannover, Welfengarten 1, 30167Hannover, Germany, [email protected] II) If dim A = 14 , then B is Morita equivalent to one of the following block algebras:(a) F D ( p = 7 ).(b) the principal -block of S with defect .(c) the principal -block of PSU(3 , with defect .(d) the principal -block of PSL(2 , with defect . The bulk of the proof is devoted to the non-existence of a certain block with extraspecial defect groupof order and exponent . The methods are quite different from those in [12]. For some of the Brauertree algebras occurring in [11] no concrete block algebra was given. For future reference we provideexplicit examples in the following table. Here, B and B denote the principal block and a suitablenon-principal block respectively. dim( A ) D Morita classes ≤ | D | = dim( A ) F D C F S C B ( A ) C F C C B (PSL(2 , | D | = 8 F D C F C , B (PSL(2 , C × C F [ C × C ] , B (2 . ( S × S ))10 C F D C B (PSL(2 , C B (PSL(2 , D F S C F C C B (PSL(2 , C × C F A C B (6 .A ) C F C , B (PSL(3 , D B (PGL(2 , SD B (3 .M ) C B (PSL(2 , C B ( S ) C F D , B (PSU(3 , C B (PSL(2 , For basic algebras of dimension there are still only finitely many corresponding Morita equivalencesclasses of blocks, but we do not know if a certain Brauer tree algebra actually occurs as a block. Thedetails are described in the last section of this paper. Before we start the proof of Theorem 1, we introduce a number of tools some of which were alreadyapplied in [11]. For more detailed definitions we refer the reader to [17].2robably the most important Morita invariant of a block B is the Cartan matrix C . It is a non-negative,integral, symmetric, positive definite and indecomposable matrix of size l ( B ) × l ( B ) where l ( B ) denotesthe number of simple modules of B . Since the simple modules of a basic algebra are -dimensional, thesum of the entries of C equals dim A in the situation of Theorem 1. The largest elementary divisor of C is the order of a defect group D of B and therefore a power of p . In particular, | D | is bounded in termsof dim A . Another Morita invariant is the isomorphism type of the center Z( B ) of B . In particular, k ( B ) := dim Z( B ) = dim Z( A ) ≤ dim A in the situation of Theorem 1.Since we encounter many blocks of defect in the sequel, it seems reasonable to construct them first. Proposition 2.
Let B be a p -block of a finite group with defect and basic algebra A . Then m :=( p − /l ( B ) is an integer, called the multiplicity of B . If l ( B ) = 1 , then dim A = p . If l ( B ) = 2 ,then dim A ∈ { p, m + 5 } . If l ( B ) = 3 , then dim A ∈ { p, m + 9 , m + 11 , m + 6 } . Moreover, if dim A ∈ { , } , then only the blocks in Theorem 1 occur up to Morita equivalence.Proof. By the Brauer–Dade theory, B is determined up to Morita equivalence by a planarly embedded Brauer tree , the multiplicity m and the position of the so-called exceptional vertex if m > . For precisedefinitions we refer to [14, Chapter 11]. If l ( B ) = 1 , then B has Cartan matrix ( p ) and the result follows(the Brauer tree has only two vertices). Now we construct the Brauer trees and Cartan matrices for l ( B ) ∈ { , } . The exceptional vertex is depicted by the black dot (if m > ).(i) C = (cid:18) m + 1 mm m + 1 (cid:19) dim A = 4 m + 2 = 2 p. This case occurs for B = F D p = A . If p = 7 , we get dim A = 14 .(ii) C = (cid:18) m + 1 11 2 (cid:19) dim A = m + 5 = p + 92 . This case occurs for the principal block of
PSL(2 , q ) whenever p divides q + 1 exactly once (see [2,Section 8.4.3]). By Dirichlet’s Theorem there always exists a prime q ≡ − p (mod p ) whichdoes the job. Choosing ( p, q ) ∈ { (17 , , (19 , } yields blocks with dim A = 13 and dim A = 14 respectively.(iii) C = m + 1 m mm m + 1 mm m m + 1 dim A = 9 m + 3 = 3 p. This case occurs for B = F [ C p ⋊ C ] = A . Obviously, there are no such blocks with dim A ∈{ , } .(iv) C = m + 1 1 11 2 11 1 2 dim A = m + 11 = p + 323 .
3e do not know if this tree always occurs as a block algebra, but it does for a non-principal -block of the -fold cover .A (see [19]). This gives an example with dim A = 13 . Obviously, dim A = 14 cannot occur here.(v) C = m + 1 1 01 2 10 1 2 dim A = m + 9 = p + 263 . By [13, Proposition 2.1], there exists a prime q such that p divides q − exactly once. Thenthe principal block of GL(3 , q ) has this form by Fong–Srinivasan [6]. The principal -block of PSL(3 , is an example with dim A = 13 . Again, dim A = 14 is impossible here.(vi) C = m + 1 m m m + 1 10 1 2 dim A = 4 m + 6 = 4 p + 143 . Again by [13, Theorem 1], there exists a prime q such that the principal block of GU(3 , q ) hasthis form. The principal -block of PSU(3 , is an example with dim A = 14 . On the other hand, dim A = 13 cannot occur.Finally, if l ( B ) ≥ , then the trace of C is ≥ and we need at least six positive off-diagonal entriesto ensure that C is symmetric and indecomposable. Hence, dim A ≤ can only occur if l ( B ) = 4 , dim A = 14 , m = 1 and the Brauer tree is a line. This happens for the principal -block of S .In order to investigate blocks of larger defect, we develop some more advanced methods. The decompo-sition matrix Q = Q of B is non-negative, integral and indecomposable of size k ( B ) × l ( B ) such that Q t Q = C . Given dim A , there are only finitely many choices for Q . Richard Brauer has introduced theso-called contribution matrix M = M := | D | QC − Q t ∈ Z k ( B ) × k ( B ) . The heights of the irreducible characters of B are encoded in the p -adic valuation of M (see [17,Proposition 1.36]). As usual, we denote the number of irreducible characters of B of height h ≥ by k h ( B ) . If k ( B ) < k ( B ) , then D is non-abelian according to Kessar–Malle’s [10] solution of one half ofBrauer’s height zero conjecture.The -blocks occurring in Theorem 1 are determined by the following proposition. Proposition 3.
Let B be a block of a finite group with Cartan matrix C = (cid:0) (cid:1) . Then B is Moritaequivalent to the principal -block of PGL(2 , or to a non-principal block of .M . Moreover, thereis no block with Cartan matrix or . roof. All three matrices have largest elementary divisor . Therefore, p = 2 and a defect group D of B has order . For the first matrix, the possible decomposition matrices are . . . . . , . . . . . The diagonal of the contribution matrix M is (5 , , , , , , or (13 , , , , . It follows that k ( B ) =4 (the first four characters have height ). By [17, Theorem 13.6], the Alperin–McKay Conjecture holdsfor all -blocks of defect . Thus, k ( B D ) = 4 where B D is the Brauer correspondent of B in N G ( D ) .Now B D dominates a block B D of N G ( D ) /D ′ with abelian defect group D/D ′ . By [14, Theorem 9.23],we conclude that k ( B D ) = k ( B D ) ≤ k ( B D ) = 4 . Now [17, Proposition 1.31] implies | D/D ′ | = 4 . Hence, D is a dihedral group, a semidihedral groupor a quaternion group. A look at [17, Theorem 8.1] (the Cartan matrices in (5a) and (5b) are mixedup) tells us that k ( B ) = 7 and D ∈ { D , SD } . The corresponding Morita equivalence classes werecomputed by Erdmann [5] (see [9, Appendix] for a definite list). Only the two stated examples occurup to Morita equivalence.For the second matrix there is only one possible decomposition matrix and we obtain similarly that k ( B ) = 4 and k ( B ) = 7 . By [17, Theorem 8.1], D ∼ = D . However, it can be seen from [9, Appendix]that there are no such blocks (all Cartan invariants are positive). Nevertheless, C occurs as Cartanmatrix with respect to a suitable basic set (for the principal block of PSL(2 , , for instance).In the last case there are two feasible decomposition matrices: . . . . .. . . , . . . . . . . . . . .. . . . The first matrix leads to k ( B ) = 4 and k ( B ) = 5 . This contradicts [17, Theorem 8.1]. The secondmatrix reveals k ( B ) = k ( B ) = 8 . Since Brauer’s height zero conjecture holds for B by [17, Theo-rem 13.6], D is abelian. By [17, Theorem 8.3], D is not isomorphic to C × C . In fact, D must beelementary abelian by [18, Proposition 16], for instance. By Eaton’s classification [3], B should beMorita equivalent to the group algebra of the Frobenius group D ⋊ C . But this is a basic algebra ofdimension .The local structure of B is determined by a fusion system F on D (again there are only finitelymany choices for F when dim A is fixed). The p ′ -group E := Out F ( D ) is called the inertial quotient of B . Recall that for every S ≤ D there is exactly one subpair ( S, b S ) attached to F (here, b S is aBrauer correspondent of B in C G ( S ) ). After F -conjugation, we may and will always assume that S is fully F -normalized . Then b S has defect group C D ( S ) and fusion system C F ( S ) . Moreover, the Brauer5orrespondent B S := b N G ( S,b S ) S has defect group N D ( S ) and fusion system N F ( S ) . If S = h u i is cyclic,we call ( u, b u ) := ( S, b S ) a subsection .Let R be a set of representatives for the F -conjugacy classes of elements in D . Then a formula ofBrauer asserts that k ( B ) = X u ∈R l ( b u ) . Each b u dominates a block b u of C G ( u ) / h u i with defect group C D ( u ) / h u i and fusion system C F ( u ) / h u i .If C u is the Cartan matrix of b u , then C u := |h u i| C u is the Cartan matrix of b u . Let Q u := ( d uχϕ : χ ∈ Irr( B ) , ϕ ∈ IBr( b u )) be the generalized decomposition matrix with respect to ( u, b u ) . The orthogonalityrelations assert that Q t u Q v = δ uv C u for u, v ∈ R where δ uv is the Kronecker delta and Q v is thecomplex conjugate of Q v . As above, we define the contribution matrices M u for each u ∈ R . Sincethe generalized decomposition numbers are algebraic integers, we may express Q u with respect to asuitable integral basis. This yields “fake” decomposition matrices e Q u which obey similar orthogonalityrelations (see [1, Section 4] for details). We call e C u := e Q t u e Q u the “fake” Cartan matrix of b u .The following curious result might be of independent interest. Proposition 4.
Let B be a p -block of a finite group with abelian defect group D and inertial quotient E .(i) If p = 2 , then l ( B ) ≡ | E | ≡ k ( E ) (mod 8) .(ii) If p = 3 , then l ( B ) ≡ | E | ≡ k ( E ) (mod 3) .Proof. We argue by induction on | D | . If | D | ≤ , then l ( B ) = | E | = k ( E ) . Thus, let | D | ≥ . Let d := 8 if p = 2 and d := 3 if p = 3 . Let R be a set of representatives for the E -orbits on D . Since E isa p ′ -group, we have | C E ( u ) | ≡ d ) for all u ∈ D . Hence, | E | X u ∈R | C E ( u ) | = X u ∈ D | C E ( u ) | ≡ | D | ≡ d ) . By Kessar–Malle [10] and [17, Proposition 1.31], k ( B ) = k ( B ) ≡ d ) . Using Brauer’s formulaand induction yields l ( B ) = k ( B ) − X u ∈R\{ } l ( b u ) ≡ − X u ∈R\{ } | C E ( u ) | ≡ | E | ≡ X χ ∈ Irr( E ) χ (1) ≡ k ( E ) (mod d ) . For the principal block B , Alperin’s weight conjecture asserts that l ( B ) = k ( E ) in the situation ofProposition 4.Finally, we study the elementary divisors of C via the theory of lower defect groups . The -multiplicity m (1) B ( S ) of a subgroup S ≤ D is defined as the dimension of a certain section of Z( B ) (the precisedefinition in [17, Section 1.8] is not needed here). Since we are only interested in -multiplicities, weomit the exponent (1) from now on. Furthermore, it is desirable to attached a multiplicity to a subpair ( S, b S ) instead of a subgroup. We do so by setting m B ( S, b S ) := m B S ( S ) . Note that ( S, b S ) is also a subpair for B S and m B S ( S, b S ) = m B ( S, b S ) . Now the multiplicity of anelementary divisor d of C is m ( d ) = X m B ( S, b S ) ( S, b S ) runs through the F -conjugacy classes of subpairs with | S | = d . In particular, m B ( D, b D ) = m ( | D | ) = 1 .We are now in a position to investigate blocks with extraspecial defect group D ∼ = 3 of order and exponent . The partial results on these blocks obtained by Hendren [8] are not sufficient for ourpurpose. We proceed in four stages. The first lemma is analogous to [17, Lemma 13.3]. Lemma 5.
Let B be a block of a finite group G with defect group D ∼ = C × C and inertial quotient E ∼ = C × C . Suppose that l ( B ) = 4 . Let D = S × T with E -invariant subgroups S ∼ = T ∼ = C . Then m B ( S, b S ) = m B ( T, b T ) = 1 .Proof. By [1, Theorem 3], B is perfectly isometric to its Brauer correspondent in N G ( D ) . It followsthat the elementary divisors of the Cartan matrix of B are , , , . In particular, m (3) = 2 . Let U ≤ D be of order such that S = U = T . Then b U is nilpotent and l ( b U ) = 1 . Since B U has defect group D ,we obtain m B U ( D ) = 1 . Hence, [17, Lemma 1.43] implies m B ( U, b U ) = m B U ( U ) = 0 . It follows that m B ( S, b S ) + m B ( T, b T ) = m (3) = 2 . (2.1)Similarly, b S has defect group D and inertial quotient C . Hence, l ( b S ) = 2 by [1, Theorem 3]. Thistime [17, Lemma 1.43] gives m B ( S, b S ) = m B S ( S ) + m B S ( D ) − ≤ l ( b S ) − and similarly, m B ( T, b T ) ≤ . By (2.1), we must have equality.Recall that every ′ -automorphism group E of D ∼ = 3 acts faithfully on D/ Φ( D ) ∼ = C × C . Thisallows us to regard E as a subgroup of the semilinear group ΓL(1 , ≤ GL(2 , . Note that ΓL(1 , isisomorphic to the semidihedral group SD . Moreover, C E (Z( D )) = E ∩ SL(2 , ≤ Q . Lemma 6.
Let B be a block of a finite group G with defect group D ∼ = 3 and inertial quotient E ∼ = SD . Suppose that Z := Z( D ) E G and that IBr( b Z ) contains at least four Brauer characterswhich are not G -invariant. Then m B ( Z, b Z ) > .Proof. Since C E ( Z ) ∼ = Q acts regularly on D/Z , there are two subgroups, say Z and S , of order in D up to F -conjugation. Hence, m (3) = m B ( Z, b Z ) + m B ( S, b S ) . We observe that B S has defect group N D ( S ) = SZ ∼ = C × C and inertial quotient C × C . By [1, Theorem 3], l ( B S ) ∈ { , } . In thefirst case, m B ( S, b S ) = 0 by [17, Lemma 1.43] and in the second case m B ( S, b S ) = m B S ( S, b S ) = 1 byLemma 5. Thus, it suffices to show that m (3) ≥ .Since E acts non-trivially on Z , we have | G : N | = 2 where N := C G ( Z ) . As usual, b Z dominatesa block b Z with defect group D/Z ∼ = C × C and inertial quotient C E ( Z ) ∼ = Q . By hypothesis, l ( b Z ) ≥ . By [1, Lemma 13], there exists a basic set Γ for b Z (which is a basic set for b Z as well) suchthat G acts on Γ and the Cartan matrix of b Z with respect to Γ is or δ ij ) i,j =1 . We may assume that θ , . . . , θ ∈ Γ such that ϕ := θ G = θ G and µ := θ G = θ G belong to a basic set ∆ of B . In order to determine the Cartan matrix C of B with respect to ∆ , we introduce the projective7ndecomposable characters Φ ϕ and Φ µ (note that these are generalized characters in our setting). By[14, Theorem 8.10], Φ ϕ = Φ Gθ and Φ µ = Φ Gθ . In particular, Φ ϕ and Φ µ vanish outside N . We compute [Φ ϕ , Φ ϕ ] = 1 | G | X g ∈ G | Φ ϕ ( g ) | = 12 1 | N | X g ∈ N | Φ ϕ ( g ) | = 12 [Φ θ + Φ θ , Φ θ + Φ θ ] = 9 = [Φ µ , Φ µ ] , [Φ ϕ , Φ µ ] = 12 [Φ θ + Φ θ , Φ θ + Φ θ ] = 6 . Let τ ∈ ∆ \ { ϕ, µ } . If τ N is the sum of two characters in Γ , then l ( b Z ) = 8 and [Φ ϕ , Φ τ ] = 6 = [Φ µ , Φ τ ] . If, on the other hand, τ N ∈ Γ , then also (Φ τ ) N = Φ τ N by [14, Corollary 8.8]. In this case we compute [Φ ϕ , Φ τ ] = [Φ µ , Φ τ ] ∈ { , } depending on l ( b Z ) . In any case, C has the form C = a · · · a s a · · · a s a a ∗ · · · ∗ ... ... ... ... a s a s ∗ · · · ∗ with a , . . . , a s ∈ { , } . By the Gauss algorithm there exist X, Y ∈ GL( l ( B ) , Z ) such that XCY = . .. .. . ∗ . Since all elementary divisors of C are powers of , it follows that m (3) ≥ as desired. Lemma 7.
Let B be a block of a finite group G with defect group D ∼ = 3 and fusion system F = F ( J ) . Then B cannot have Cartan matrix (cid:0) (cid:1) .Proof. By way of contradiction, suppose that B has the given Cartan matrix C . Then B has decom-position matrix . . .. . .
11 1 or . . . . . .. . .
11 1 . The diagonal of the contribution matrix M is (16 , , , , , , or (4 , , , , , , , , , . It followsthat k ( B ) ∈ { , } and k ( B ) = 1 (the last row corresponds to the character of height ). Fromthe Atlas we know that all non-trivial elements of D are F -conjugate. Let ( z, b z ) be a non-trivial8ubsection such that z ∈ Z := Z( D ) . By [16, Table 1.2], B has inertial quotient SD . It follows that b z is a block with defect group D and inertial quotient Q . Moreover, l ( b z ) = k ( B ) − l ( B ) ∈ { , } .The possible Cartan matrices of b z are given in the proof of Lemma 6. The generalized decompositionnumbers d zχϕ are Eisenstein integers and can be expressed with respect to the integral basis , e πi/ .According to the action of N G ( Z, b z ) on IBr( b z ) there are eight possibilities for the “fake” Cartanmatrix e C z which are listed explicitly in [1, proof of Lemma 14]. In each case we apply an algorithmof Plesken [15] (implemented in GAP [7]) to determine the feasible “fake” decomposition matrices e Q z . To this end we also take into account that the diagonal of M z is (11 , , , , , , or (23 , , , , , , , , , , since M + M z = | D | k ( B ) . It turns out that only two of the eightcases can actually occur. If k ( B ) = 7 , then N G ( Z, b z ) has one fixed point in IBr( b z ) and if k ( B ) = 10 ,then N G ( Z, b z ) has two fixed points in IBr( b z ) . Hence, in both cases the block B Z fulfills the assumptionof Lemma 6. Consequently, m (3) = m B ( Z, b Z ) = m B Z ( Z, b Z ) > . However, the elementary divisors of C are and . Contradiction. Proposition 8.
There does not exist a block of a finite group with Cartan matrix (cid:0) (cid:1) .Proof.
As in Lemma 7, any block B with the given Cartan matrix C has a defect group D of order . The possible decomposition matrices were also computed in the proof of Lemma 7. In particular, k ( B ) ∈ { , } , k ( B ) = 1 and k ( B ) − l ( B ) ∈ { , } . By Kessar–Malle [10], D is nonabelian. By [17,Theorem 8.15], D cannot have exponent , i. e. D ∼ = 3 . The fusion systems F on that group wereclassified in Ruiz–Viruel [16]. As explained before, we regard the inertial quotient E of B as a subgroupof SD . Let R be a set of representatives for the F -conjugacy classes in D . For = u ∈ R we have l ( b u ) ≡ | C E ( u ) | (mod 3) by Proposition 4 (applied to b u if u ∈ Z := Z( D ) ). Therefore, the residue of k ( B ) − l ( B ) modulo only depends on F . If D contains F -essential subgroups, then F is the fusionsystem of one of the following groups H : C ⋊ SL(2 , , C ⋊ GL(2 , , PSL(3 , , PSL(3 , . , F (2) ′ , J . The last case was excluded in Lemma 7. In the remaining cases we can compare with the principalblock of H to derive the contradiction ≡ k ( B ) − l ( B ) ≡ k ( B ( H )) − l ( B ( H )) . Hence, there are no F -essential subgroups, i. e. F = F ( D ⋊ E ) . Suppose that E ≤ Q . Then N G ( Z, b Z ) =C G ( Z ) and b Z = B Z has fusion system F as well. If E = 1 , then B is nilpotent in contradiction to l ( B ) = 2 . Thus, let E = 1 . Let B Z be the block with defect group D/Z dominated by B Z . By [1, The-orem 3] and Proposition 4, l ( B Z ) = l ( B Z ) ≥ . Since E acts semiregularly on D/Z , the Cartan matrixof B Z has elementary divisors and (see [17, Proposition 1.46]). Hence, is an elementary divisor ofthe Cartan matrix of B Z . Since Z ≤ Z(C G ( Z )) , it follows that m (3) ≥ m B ( Z, b Z ) = m B Z ( Z, b Z ) > by [17, Lemma 1.44]. A contradiction.We are left with the situation E * Q . Here, R ∩ Z = { , z } . The case E ∼ = C × C is impossibleby a comparison with the principal block of D ⋊ E as above. We summarize the remaining cases (thesecond column refers to the small groups library in GAP): E realizing group l ( b z ) P u/ ∈ Z( D ) l ( b u ) C
54 : 5 1 2 + 2 + 1 + 1 + 1 C
216 : 86 4 1 D
216 : 87 4 2 + 2
9n the first case we have k ( B ) = 9 and there exists u ∈ R \ Z such that u and u − are F -conjugate.Then l ( b u ) = 1 and the Cartan matrix of b u is (9) . The generalized decomposition matrix Q u is integral,since Q u = Q u − = Q u . The only choice up to signs is Q u = ( ± , . . . , ± , t where the last characterhas height . However, Q u cannot be orthogonal to the decomposition matrix of B as computed inLemma 7. Next let E ∼ = C . Here k ( B ) = 6 and the generalized decomposition matrix Q u has theform Q u = ( ± , ± , . . . , ± , t . More precisely, Q and Q u can be arranged as follows ( Q , Q u ) = . . − . − . . − . −
11 1 . . From that we compute the diagonal of the contribution matrix M z as (8 , , , , , , . By [1,Proposition 7] (applied to the dominated block with defect group C × C ), there exists a basic set Γ for b z such that the Cartan matrix becomes δ ij ) i,j =1 and N G ( Z, b Z ) acts on Γ . There are threesuch actions. In each case we may compute the “fake” Cartan matrix e C z and apply Plesken’s algorithm.It turns out that none of those cases leads to a valid configuration.Finally, let E ∼ = D and u ∈ R such that l ( b u ) = 2 . We check that u is F -conjugate to u − . The Cartanmatrix of b u is (cid:0) (cid:1) up to basic sets. Let U := h u i . If N G ( U, b u ) interchanges the Brauer charactersof b u , then the “fake” Cartan matrix becomes e C u = (cid:0) (cid:1) (see [1, proof of Lemma 14], for instance).But then k ( B ) ≤ which is not the case. Therefore, N G ( U, b u ) fixes the Brauer characters of b u and B U satisfies l ( B U ) = 4 by Clifford theory. By Lemma 5, we conclude that m (3) ≥ m B ( U, b U ) = m B U ( U, b U ) = 1 . This is the final contradiction.Since the contribution matrix does not depend on basic sets, the proof shows more generally that (cid:0) (cid:1) cannot be the Cartan matrix of a block with respect to any basic set. This is in contrast to the mainresult of [12] where the authors showed that (cid:0) (cid:1) is not the Cartan matrix of a block with defectgroup C × C , although a transformation of basic sets results in the Cartan matrix (cid:18) (cid:19) = (cid:18) − (cid:19) (cid:18) (cid:19) (cid:18) − (cid:19) of the Frobenius group C ⋊ C . Suppose that B is a block with basic algebra A of dimension and Cartan matrix C . We discuss thevarious possibilities for C . If l ( B ) = 1 , then C = (13) , p = 13 and B has defect . This is coveredby Proposition 2. For l ( B ) = 2 we obtain the following possibilities for C up to labeling of the simplemodules: C (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) det C
17 23 27 29 10 14 16 3 and are not prime powers. If det C is a prime, then the result follows fromProposition 2. The remaining cases det C ∈ { , } were handled in Proposition 3 and Proposition 8respectively.Now we turn to l ( B ) = 3 . Up to labeling, the following possibilities may arise: C det C
16 13 19 18 17 C det C
21 7 1 2
The determinants , and are impossible and the prime determinants are settled by Proposition 2.The remaining case was done in Proposition 3.If l ( B ) ≥ , then the trace of C is ≥ . Since C is symmetric and indecomposable, we need at least sixmore non-zero entries. But then dim A ≥ . In this section, B is a block with basic algebra A of dimension . Since is not a prime power, l ( B ) ≥ . In view of Proposition 2, we only list the possible Cartan matrices C such that det C is aprime power, but not a prime: C det C
16 25 4 4 4
The -blocks of defect were classified by Erdmann [5]. The Morita equivalence classes are representedby F D , F A and B ( A ) . Only the last block did not already appear in Linckelmann’s list. It is easy tocheck that B ( A ) has a basic algebra of dimension . The case det C = 16 was done in Proposition 3.Now let det C = 25 and p = 5 . Since l ( B ) = 3 does not divide p − , D is elementary abelian oforder . The decomposition matrix is . In particular, k ( B ) − l ( B ) = 5 . Let E ≤ GL(2 , be the inertial quotient of B . Every non-trivialsubsection ( u, b u ) satisfies l ( b u ) = | C E ( u ) | by Brauer–Dade. In particular, k ( B ) − l ( B ) only depends11n the action of E on D . An inspection of [1, Theorem 5] shows that k ( B ) − l ( B ) = 5 never occurs.Hence, this case is impossible as well. While classifying blocks B with basic algebra of dimension , only the following Cartan matrices arehard to deal with: , . The first matrix belongs to a Brauer tree algebra and could potentially arise from a -block of defect (see Proposition 2). David Craven has informed me that such a block does most likely not exist(assuming the classification of finite simple groups).The second matrix leads, once again, to a defect group D of order . Moreover, k ( B ) = k ( B ) ∈ { , } .Arguing along the lines of Proposition 8, it can be shown with some effort that D is abelian. Now theblock is ruled out by Proposition 4.Finally, for basic algebras of dimension , a × Cartan matrix with largest elementary divisor shows up. We made no attempt to say something about such blocks. Acknowledgment
I thank David Craven for providing detailed information on the possible trees of block algebras. Thiswork is supported by the German Research Foundation (SA 2864/1-2 and SA 2864/3-1).
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