James's Conjecture holds for blocks of q-Schur algebras of weights 3 and 4
JJAMES’S CONJECTURE HOLDS FOR BLOCKS OF q -SCHUR ALGEBRAS OFWEIGHTS 3 AND 4 AARON YI RUI LOW
Abstract.
When the characteristic of the underlying field is at least 5, we prove that the adjust-ment matrix for blocks of q -Schur algebras of weights 3 and 4 is the identity matrix. Moreover, weshow that the decomposition numbers for weight 3 blocks of q -Schur algebras are bounded aboveby one. Introduction
Given a positive integer m , let N m> := { ( a , a , . . . , a m ) | a i ∈ N > ∀ ≤ i ≤ m } be the set of m -tuples of positive integers. We now fix an n ∈ N > . Definition 1.1. An m -tuple λ = ( λ , λ , . . . , λ m ) ∈ N m> is a composition of n if m (cid:80) i =1 λ i = n ; ifmoreover, λ i ≥ λ i +1 ∀ ≤ i ≤ m −
1, we say that λ is a partition of n . We refer to λ i as the parts of λ and denote the number of parts of λ by l ( λ ) := m .We denote the set of all partitions of n by P ( n ). Definition 1.2.
Let e ≥ λ e-singular if there is an integer1 ≤ i ≤ l ( λ ) − e + 1 such that λ i = λ i +1 = · · · = λ i + e − (cid:54) = 0; we call λ e-regular otherwise. Wedenote the set of all e -singular partitions of n by P e -sing ( n ) and the set of all e -regular partitionsof n by P e -reg ( n ).Suppose that q is a non-zero element of a field F . Unless mentioned explicitly, we assumethat F can have any characteristic (including zero). Let e be the least positive integer such that1 + q + · · · + q e − = 0, assuming throughout the paper that it exists. Let n be a positive integer.Denote by S n = S q ( n, n ) the q-Schur algebra (over F ) defined in [5]. When q = 1, this is justthe classical Schur algebra over F . To each partition λ ∈ P ( n ), we associate a Weyl module W λ . Each W λ has a simple head L λ which is self-dual with respect to the contravariant dualityinduced by the anti-automorphism of S n . Moreover, the set { L λ | λ ∈ P ( n ) } is a complete setof mutually non-isomorphic simple modules of S n . Given any two partitions λ, µ ∈ P ( n ), thecomposition multiplicities [ W λ : L µ ] are called the decomposition numbers of S n . We record thesein a decomposition matrix with rows and columns indexed by P ( n ) and whose ( λ, µ )-entry is[ W λ : L µ ].Let H n = H F ,q ( S n ) denote the Iwahori-Hecke algebra of the symmetric group S n . This is a‘deformation’ of the group algebra F S n ; we refer the reader to [17] for its definition. When q = 1,this is simply the group algebra F S n . The representation theory of H n is very similar to that of S n .To each partition λ ∈ P ( n ), we associate a Specht Module S λ for H n . If λ ∈ P e -reg ( n ), then S λ has asimple head D λ . The set { D λ | λ ∈ P e -reg ( n ) } is a complete set of mutually non-isomorphic simple H n -modules. The decomposition numbers of H n are the composition multiplicities [ S λ : D µ ]; thedecomposition matrix for H n has rows indexed by P ( n ) and columns indexed by P e -reg ( n ), with( λ, µ )-entry [ S λ : D µ ]. Date : 28 January 2021. a r X i v : . [ m a t h . R T ] F e b arning: The Specht and Weyl modules, S λ and W λ defined in [17] are isomorphic to the dualof the Specht and Weyl modules defined by Dipper and James in [4] indexed by λ (cid:48) . We adopt theSpecht and Weyl modules defined by Dipper and James in [4] rather than those in [17].A central problem in the study of the representation theory of S n and H n is to determine theirdecomposition matrices. These two problems are closely related. In fact, there is an exact functorcalled the Schur functor from the category of S n -modules to the category of H n -modules. Given µ ∈ P ( n ), the Schur functor sends the Weyl module W µ to the Specht module S µ . If µ ∈ P e -reg ( n ),the simple module L µ is sent to the simple module D µ ; otherwise if µ ∈ P e -sing ( n ), L µ is sent tozero. Using the Schur functor, one can deduce the following: Theorem 1.3. [17, Theorem 4.18]
Suppose that λ and µ are partitions of n and that µ is e -regular.Then, [ W λ : L µ ] = [ S λ : D µ ] . In other words, the decomposition matrix of H n is a submatrix of the decomposition matrixof S n ; obtained by deleting the columns indexed by P e -sing ( n ). Lascoux, Leclerc and Thibonconjectured that their recursive LLT algorithm [15] calculates the decomposition matrices of H n over C . This was later proved by Ariki in [1]. It is known that decomposition matrices over fieldsof prime characteristic may be obtained from decomposition matrices over C by post-multiplyingby an ‘adjustment matrix’. In view of this, we often study the adjustment matrices instead of thedecomposition matrices directly when working over fields of prime characteristic. The complexityof the representation theory of a block of H n or S n is roughly captured by a measure called theweight of that block. James conjectured [11] that when the weight of a block is less than thecharacteristic p > H n ,the conjecture has been proved for weights up to 4 by the works of Richards [19] and Fayers [8, 9].The author [16] has also proved that the adjustment matrix for the principal block of H e of weight5 is the identity matrix when char( F ) ≥ e (cid:54) = 4. For the q -Schur algebras, the weight 2case was proved by Schroll and Tan [21]. However, Williamson found a counter-example [25] toJames’s Conjecture. Nevertheless, the smallest counter-example produced in his paper occurs inthe symmetric group S n where n = 1744860. There is considerable interest in finding smallercounter-examples.This paper is structured in the following way. In section 2, we mention some fundamental resultsin the modular representation theory of H n and S n . Our main object of interest, the adjustmentmatrices are introduced in section 3. We give an overview of the existing results on adjustmentmatrices and provide some new tools for studying them. In sections 4 and 5, we apply the resultsfrom sections 2 and 3 to prove James’s Conjecture for blocks of S n of weights 3 and 4. We alsoshow that the decomposition numbers for weight 3 blocks of S n are bounded above by one.2. Background
Partitions and abacus displays.
Take an abacus with e vertical runners, numbered 0 , . . . , e − , , . . . on the runners increasing from left to right alongsuccessive ‘rows’. Given λ ∈ P ( n ), take an integer r ≥ l ( λ ). Define β i = (cid:40) λ i + r − i, if 1 ≤ i ≤ l ( λ ) ,r − i, if r > l ( λ ) . Now, place a bead at position β i for each i . The resulting configuration is an abacus display for λ with r beads. If a bead has been placed at position j , we say that the position j is occupied .Otherwise, we say that the position j is unoccupied or vacant . We remark that moving a bead fromposition a to an unoccupied position b < a is akin to removing a rim hook of length h = b − a fromthe Young diagram of λ . The leg-length of the hook is given by the number of occupied positions etween a and b . By moving all the beads as high as possible on their runners, the resultingconfiguration is an abacus display for the e-core of λ . The relative e - sign of λ , denoted by σ e ( λ ) is( − t , where t is the total leg lengths of e -hooks removed to obtain the e -core (See [18, § t depends on the choice of e -hooks removed, its parity is constant. We define the e-weight of a bead to be the number of vacant positions above it on the same runner. The e -weightof λ is the sum of e -weights of all the beads in an abacus display for λ . Thus, if λ has e -weight w ,then its e -core is a partition of n − ew . If there is no ambiguity, we often just refer to e -weight as weight and e -core as core .It is easy to read addable and removable nodes from an abacus display. Display λ on an abacuswith e runners and r beads, where r ≥ l ( λ ). Let i be the residue class of ( j + r ) modulo e . Then,the removable nodes of λ with e -residue j correspond to the beads on runner i with unoccupiedpreceding positions, while the addable nodes of e -residue j correspond to the vacant positions onrunner i with occupied preceding positions. We call the removable (resp. addable) nodes with e -residue j j-removable (resp. j-addable ). Removing a removable node corresponds to moving thecorresponding bead to its preceding unoccupied position, while adding an addable node correspondsto moving the corresponding bead to its succeeding unoccupied position. We adopt the conventionthat position 0 has an occupied preceding position.2.2. Blocks of S n and H n .Theorem 2.1. [17, Theorem 5.37] Let λ and µ be partitions of n . Then, W λ and W µ lie in thesame block of S n if and only if λ and µ have the same e -core. Applying the Schur functor to the theorem above, we get the following corollary.
Corollary 2.2. (Nakayama Conjecture) [17, Corollary 5.38]
Let λ and µ be partitions of n . Then, S λ and S µ lie in the same block of H n if and only if λ and µ have the same e -core. Given a block B of S n or H n , we say that a partition λ ∈ P ( n ) lies in B if W λ or S λ lies in B .Given the Nakayama Conjecture, we may define the e -weight and e -core of a block B of S n or H n simply to be the e -weight and e -core of a partition lying in B .Let λ ( i ) be the partition corresponding to the abacus display containing only a single runner,the i th runner. Denote the number of beads on the i th runner as b i . Then, we may write λ as (cid:104) λ (0) , . . . , ( e − λ ( e − | b , . . . , b e − (cid:105) ;we omit i λ ( i ) if λ ( i ) = ∅ and write i λ ( i ) simply as i if λ ( i ) = (1). Additionally, we may omit b , . . . , b e − if there is no ambiguity. If λ has e -weight w and lies in a block B of S n or H n , we saythat B is the block of e -weight w with the (cid:104) b , . . . , b e − (cid:105) notation.2.3. Jantzen-Schaper formula and the product order.
Let (cid:68) denote the usual dominanceorder for partitions. Due to the fact that S n is a cellular algebra [10], we have the following result. Theorem 2.3. [5, Corollary 4.13]
Suppose that λ and µ are partitions of n . We have • [ W µ : L µ ] = 1 , • [ W λ : L µ ] = 0 unless µ (cid:68) λ . Let λ be a partition and consider its abacus display, say with r beads. Suppose that after movinga bead at position a up its runner to a vacant position a − ie , we obtain the partition µ . Denote l λµ for the number of occupied positions between a and a − ie , and let h λµ = i .Further, write λ µ −→ τ if the abacus display of τ with r beads is obtained from that of µ by movinga bead at position b − ie to a vacant position b , and a < b . Definition 2.4.
Jantzen-Schaper bound
Let p = char( F ). For any ordered pair ( λ, τ ), we define the Jantzen-Schaper coefficient to be the nteger J F ( λ, τ ) := (cid:88) λ σ −→ τ ( − l λσ + l τσ +1 (1 + v p ( h λσ )) , where v p denotes the standard p -valuation if p > v ( x ) = 0 ∀ x . The Jantzen-Schaper boundis defined as the integer B F ( λ, µ ) = (cid:88) τ J F ( λ, τ )[ W τ F : L µ F ] . Remark . If λ has e -weight w and p > w , then v p ( h λσ ) is always zero. In this case, J F ( λ, τ ) isindependent of char( F ) and we may just refer to it as J ( λ, τ ). Similarly, if B F ( λ, τ ) turns out to beindependent of char( F ), we just refer to it as B ( λ, τ ). Theorem 2.6.
Jantzen-Schaper formula( [13, Theorem 4.7] ) [ W λ F : L µ F ] ≤ B F ( λ, µ ) . Moreover, the left-hand side is zero if and only if the right-hand side is zero.
Corollary 2.7. If B F ( λ, µ ) ≤ , then [ W λ F : L µ F ] = B F ( λ, µ ) . We write λ → τ if there exists some µ such that λ µ −→ τ . Further, write λ < J σ if there existpartitions τ , τ , . . . , τ r such that τ = λ , τ r = σ and τ i − → τ i ∀ i ∈ { , , . . . , r } . We call ≤ J the Jantzen order and it is clear that this defines a partial order on the set of all partitions. Onlypartitions in the same block are comparable under this partial order. Moreover, the dominanceorder extends the Jantzen order in the following sense: µ > J λ implies µ (cid:66) λ . Theorem 2.6 can beused to refine Theorem 2.3 the following way: Theorem 2.8.
Suppose that λ and µ are partitions of n. Then, • [ W µ : L µ ] = 1; • [ W λ : L µ ] > ⇒ µ ≥ J λ. It is difficult to check that µ ≯ J λ by inspection. To this end, we introduce the product order on partitions which was first defined by Tan in [23]. Let λ be a partition, displayed on an abacuswith e runners and r beads. Suppose that the beads having positive e-weights are at positions a , a , . . . , a s with weights w , w , . . . , w s respectively. The induced e-sequence of λ , denoted s ( λ ) r ,is defined as s (cid:71) i =1 ( a i , a i − e, . . . , a i − ( w i − e ) , where ( b , b , . . . , b t ) (cid:116) ( c , c , . . . , c u ) denotes the weakly decreasing sequence obtained by rearrang-ing terms in the sequence ( b , . . . , b t , c , . . . , c u ). Note that s ( λ ) r ∈ N w> , where w is the e -weight of λ . We define a partial order ≥ P on the set of partitions by: µ ≥ P λ if and only if µ and λ have thesame e -core and e -weight, and s ( µ ) r ≥ s ( λ ) r (for sufficiently large r ) in the standard product orderof N w> . Lemma 2.9. ( [23, Lemma 2.9] ) λ ≤ J µ ⇒ λ ≤ P µ. Therefore, µ (cid:3) P λ ⇒ [ W λ : L µ ] = 0. xample 2.10. Suppose that e = 5, r = 10, λ = (cid:104) , | (cid:105) = (10 , , , , ) and µ = (cid:104) , , | (cid:105) = (15 , , ). λ µ Then, s ( λ ) = (19 , , , , s ( µ ) = (24 , , , , µ (cid:66) λ but µ ≯ P λ .2.4. v -decomposition numbers. For a brief introduction to v-decomposition numbers , the readermay refer to [21, § λ and µ of n , we may define a polynomial d eλµ ( v ) ∈ N [ v ] with the following crucial property (which explainsits name): Theorem 2.11.
Let λ, µ ∈ P ( n ) . Then,(1) d eλµ (1) = [ W λ C : L µ C ] ,(2) ddv ( d eλµ ( v )) | v =1 = B C ( λ, µ ) .Proof. (1) has been proven by Varagnolo and Vasserot in [24]. (2) was proved by Schroll and Tanin [21, Theorem 2.13]. (cid:3) The following result tells us that d eλµ ( v ) is either an even or an odd polynomial, depending onthe relative e -signs of λ and µ . Theorem 2.12. [22, Theorem 2.4] If d eλµ ( v ) (cid:54) = 0 , then d eλµ ( v ) ∈ (cid:40) N [ v ] if σ e ( λ ) = σ e ( µ ) ,v N [ v ] otherwise . Remark . In section 5, we will sometimes calculate that B C ( λ, µ ) = 2 for some pair of partitions( λ, µ ). In order to verify whether [ W λ C : L µ C ] is 1 or 2, we would calculate σ e ( λ ) and σ e ( µ ). If theyturn out to be the same, then d eλµ ( v ) = v and [ W λ C : L µ C ] = 1 by Theorem 2.11 and Theorem 2.12.Otherwise, d eλµ ( v ) = 2 v and [ W λ C : L µ C ] = 2.The following two theorems commonly known as the Runner Removal Theorems allow us torelate v -decomposition numbers for different values of e . Theorem 2.14. [14, Theorem 4.5]
Suppose that e ≥ . Let λ and µ be partitions lying in the sameblock, and display them on an abacus with e runners and r beads, for some large enough r . Supposethat there exists some i such that in both abacus displays, the last bead on runner i occurs beforeevery unoccupied space on the abacus. Define two abacus displays with e − runners by deletingrunner i from each display, and let λ − and µ − be the partitions defined by these displays. Then, d eλµ ( v ) = d e − λ − µ − ( v ) . Theorem 2.15. [7]
Suppose that e ≥ . Let λ and µ be partitions lying in the same block, anddisplay them on an abacus with e runners and r beads, for some large enough r . Suppose that thereexists some i such that in both abacus displays, the first unoccupied space on runner i occurs afterevery bead on the abacus. Define two abacus displays with e − runners by deleting runner i fromeach display, and let ˆ λ and ˆ µ be the partitions defined by these displays. Then, d eλµ ( v ) = d e − λ ˆ µ ( v ) . emark . Calculating v -decomposition numbers in practice. When µ ∈ P e -reg ( n ), wehave a relatively fast recursive algorithm for calculating d eλµ ( v µ ∈ P e -sing ( n ), we may calculate d eλµ ( v ) by adding an empty runner to λ and µ and usingTheorem 2.14. By Theorem 2.11, the decomposition matrix for S n can be calculated in principlewhen the underlying field is C . Example 2.17.
Suppose that e ≥ µ = (cid:104) , , , | e (cid:105) , λ = (cid:104) , | e (cid:105) , ˆ µ = (cid:104) , , , | (cid:105) ,ˆ λ = (cid:104) , | (cid:105) , µ + = (cid:104) , , , | , (cid:105) and λ + = (cid:104) , | , (cid:105) . µ ˆ µ µ + λ ˆ λ λ + By Theorem 2.15, d eλµ ( v ) = d λ ˆ µ ( v ). By Theorem 2.14 and the LLT algorithm, d λ ˆ µ ( v ) = d λ + µ + ( v ) = v .2.5. The modular branching rules.
We use some notational conventions for modules. We write M ∼ M a + M a + · · · + M a t t to indicate that M has a filtration in which the factors are M , . . . , M t appearing a , . . . , a t timesrespectively. Additionally, we write M ⊕ a to indicate the direct sum of a isomorphic copies of M .There are restriction and induction functors which are exact functors between S n − and S n . If M is a module for S n , the restriction of M to S n − t is denoted by M ↓ S n − t . Similarly, the induction of M to S n + t is denoted by M ↑ S n + t . If B is a block of S n − t , we write M ↓ B to indicate the projectionof M ↓ S n − t onto B . Similarly, if C is a block of S n + t , we write M ↑ B to indicate the projection of M ↑ S n + t onto C . In this section, we describe the restriction and induction of Weyl modules andsimple modules.Suppose that A , B and C are blocks of S n − k , S n and S n + k respectively, and that there is an e -residue j such that a partition lying in A may be obtained from a partition lying in B by removingexactly k j -removable nodes, while a partition lying in C may be obtained from a partition lyingin B by adding exactly k j -addable nodes.Suppose that λ is a partition in B , and that λ − , λ − , . . . , λ − t are the partitions in A that may beobtained from λ by removing k j -removable nodes. Similarly, let λ +1 , λ +2 , . . . , λ + s be the partitionsin C that may be obtained from λ by adding k j -addable nodes. We have the following result. Theorem 2.18. (The Branching Rule [2] ) Suppose that A , B , C and λ are as above. Then, W λ ↓ A ∼ ( W λ − ) k ! + ( W λ − ) k ! + · · · + ( W λ − t ) k ! and W λ ↑ C ∼ ( W λ +1 ) k ! + ( W λ +2 ) k ! + · · · + ( W λ + s ) k ! . We now discuss the restriction and induction of simple modules. Suppose that the nodes withresidue j are on runner i . The j-signature of λ is the sequence of signs defined as follows. Startingfrom the top row of the abacus display for λ and working downwards, write a − if there is a beadon runner i but no bead on runner i −
1; write a + if there is a bead on runner i − i ; write nothing for that row otherwise. Given the j -signature of λ , successively deleteall neighbouring pairs of the form − + to obtain the reduced j-signature of λ . If there are any − resp. +) signs in the reduced j -signature of λ , we call the corresponding nodes on runner i normal (resp. conormal ). Normal (resp. conormal) nodes with residue j are also called j-normal (resp. j-conormal ). Definition 2.19.
Let λ be a partition. We denote the number of j -normal nodes of λ by (cid:15) j ( λ ) andthe number of j -conormal nodes of λ by ϕ j ( λ ). For t ≤ (cid:15) j ( λ ), we define ˜ E tj λ to be the partitionobtained from λ by removing the t highest (in an abacus display for λ ) j -normal nodes. For t ≤ ϕ j ( λ ), we define ˜ F tj λ to be the partition obtained from λ by adding the t lowest (in an abacusdisplay for λ ) j -conormal nodes. Theorem 2.20. [2]
Suppose that A , B , C and λ are as above. • If (cid:15) j ( λ ) < k , then L λ ↓ A = 0 . • If (cid:15) j ( λ ) > k , then soc ( L λ ↓ A ) ∼ = ( L ˜ E kj λ ) ⊕ k ! . • If (cid:15) j ( λ ) = k , then L λ ↓ A ∼ = ( L ˜ E kj λ ) ⊕ k ! . • If ϕ j ( λ ) < k , then L λ ↑ C = 0 . • If ϕ j ( λ ) > k , then soc ( L λ ↑ C ) ∼ = ( L ˜ F kj λ ) ⊕ k ! . • If ϕ j ( λ ) = k , then L λ ↑ C ∼ = ( L ˜ F kj λ ) ⊕ k ! . The following lemma guarantees that the weight of a partition will not increase if we remove allof its j -normal nodes (or add all of its j -conormal nodes) for some e -residue j . Lemma 2.21.
Suppose that µ lies in a block of S n of weight w . Let k = (cid:15) j ( µ ) and l = ϕ j ( µ ) .Then, ˜ E kj µ and ˜ F lj µ have weight w − kl . Before we proceed with the proof of this lemma, it may be helpful to first look at an example.
Example 2.22.
In the diagrams below, we only display two runners of the abacus displays for thepartitions; the runner on the right corresponds to the nodes with e -residue equal to j . We highlightthe reduced j -signature in red. In this example, k = 3 and l = 2. µ ˜ E kj µ ˜ F lj µ Assuming for simplicity that the other runners which are not displayed in the diagram have noweight, we may count that µ has weight 43 while ˜ E kj µ and ˜ F lj µ have weight 37. Proof.
We only show the proof for ˜ E kj µ here as the other case is similar. Let µ be a partition, k = (cid:15) j ( µ ) and l = ϕ j ( µ ). We focus our attention on the two adjacent runners, with the runner onthe right corresponding to the nodes with e -residue equal to j . Our task is to keep track of thechange of weight of each bead in these two runners when ˜ E kj is applied to µ . We may categorizethe beads in these two runners into four categories:(1) Normal; in which case there are no conormal beads in the rows below it.(2) Conormal; in which case there are no normal beads in the rows above it.(3) Two beads in the same row.
4) Two beads in two distinct rows forming a ( − +) pair that was deleted from the j -signatureto form the reduced j -signature; by definition, there are no normal or conormal beads inbetween these two rows.The change of weight of each of these types of beads when ˜ E kj is applied to µ is summarised below:(1) Each normal bead loses weight l when moved to the left.(2) The conormal beads do not experience any change in weight since there are no normal beadsabove them.(3) If two beads are in the same row and that there are α normal beads above this row, thenthe bead on the right gains weight α while the bead on the left loses weight α . Therefore,there is no net change in weight contributed by beads of this type.(4) Suppose that we have two beads in two distinct rows forming a ( − +) pair that was deletedfrom the j -signature to form the reduced j -signature. If there are β normal beads in therows above this ( − +) pair (this is well defined), then the bead on the right correspondingto the − gains weight β , while the bead on the left corresponding to the + gains weight β .Therefore, there is no net change in weight contributed by beads of this type.Since there are k normal beads, ˜ E kj µ must have weight kl less than µ . (cid:3) Adjustment Matrices
Let the q -Schur algebra over an arbitrary field F be denoted by S n , and denote the q -Schuralgebra over C by S n . Let ζ be a primitive e th -root of unity in C . We write H n for H C ,ζ ( S n ) and H n for H F ,q ( S n ). By Theorem 2.1, the Weyl modules corresponding to two partitions lie in thesame block of S n if and only if they lie in the same block of S n . Similarly, the Specht modulescorresponding to two partitions lie in the same block of H n if and only if they lie in the sameblock of H n by Corollary 2.2. Therefore, given a block B of S n or H n , we may denote B to be itscorresponding block in S n or H n respectively.The Grothendieck group G ( S n ) of S n is the additive abelian group (with complex coefficients)generated by the symbols [ E ], where E runs over the isomorphism classes of finite dimensional S n -modules, together with the relations [ F ] = [ E ] + [ G ] whenever there exists a short exact sequence0 → E → F → G →
0. Thus, as a complex vector space, G ( S n ) has a basis given by { [ L λ F ] | λ ∈P ( n ) } . We denote the decomposition matrix for S n by D S . Since D S is unitriangular, { [ W λ F | λ ∈P ( n ) } must be another basis for G ( S n ) with D S being the transition matrix between these twobases. In other words, given any λ ∈ P ( n ),[ W λ F ] = (cid:88) µ ∈P ( n ) [ W λ F : L µ F ][ L µ F ] . There is a well-defined homomorphism d S : G ( S n ) → G ( S n ) which fixes the Weyl modules, d S ([ W λ C ]) = [ W λ F ] ∀ λ ∈ P ( n ). In the literature [17], this homomorphism is known as the de-composition map . Theorem 3.1. [17, Theorem 6.35]
Let D S and D S be the decomposition matrices for S n and S n respectively. Let A S be the matrix ( a S µν ) µ,ν ∈P ( n ) , where a S µν ’s satisfy d S ([ L µ C ]) = (cid:80) ν ∈P ( n ) a S µν [ L ν F ] .Then, a S µν ∈ N for all µ, ν ∈ P ( n ) and D S = D S A S . We call the matrix A S in Theorem 3.1 the adjustment matrix for S n . Replacing S n by H n and S n by H n in Theorem 3.1 yields the following theorem. heorem 3.2. Let D H and D H be the decomposition matrices for H n and H n respectively. Let A H be the matrix ( a H µν ) µ,ν ∈P e -reg ( n ) , where a H µν ’s satisfy d H ([ D µ C ]) = (cid:80) ν ∈P e -reg ( n ) a H µν [ D ν F ] . Then, a H µν ∈ N for all µ, ν ∈ P e -reg ( n ) and D H = D H A H . We call the matrix A H in Theorem 3.2 the adjustment matrix for H n . The matrix A S has rowsand columns indexed by P ( n ), whereas A H has rows and columns indexed by P e -reg ( n ). One mayargue using the Schur functor that A H is the submatrix of A S obtained by removing the rows andcolumns of A S indexed by P e -sing ( n ). Therefore, we refer to adjustment matrices as A when it isclear whether we are dealing with S n or H n . Moreover, given any two partitions λ, µ ∈ P ( n ), wemay refer to the ( λ, µ )-entry of A as adj λµ without any ambiguity.We highlight the unitriangular property of adjustment matrices inherited from the unitriangu-larity of the decomposition matrices in the following corollary. Corollary 3.3.
Suppose that λ and µ are partitions lying in a block B of S n . Then, • adj µµ = 1 , • adj λµ = 0 unless µ ≥ J λ . It follows from Lemma 2.9 and Corollary 3.3 that µ (cid:3) P λ ⇒ adj λµ = 0. As mentioned before,it is difficult to check that µ (cid:3) J λ , whereas µ (cid:3) P λ can be verified by inspection. In terms ofadjustment matrices, we have the following corollary of Theorem 2.11. Corollary 3.4.
Suppose that λ and µ are partitions lying in a block B of S n of weight w < char( F ) .Additionally, suppose that adj νµ = 0 for all partitions ν such that λ < J ν < J µ , and that d eλµ ( v ) ∈{ , v } . Then, adj λµ = 0 . Proof.
The proof of this corollary is essentially the same as the proof of [9, Corollary 2.12] byreplacing S ν with W ν and D µ with L µ . (cid:3) In section 5, we will use the following easy fact several times.
Lemma 3.5.
Let λ and µ be two distinct partitions lying in some block B of S n . If [ W λ F : L µ F ] =[ W λ C : L µ C ] , then adj λµ = 0 .Proof. By Theorem 3.1, Theorem 2.8 and Corollary 3.3,[ W λ F : L µ F ] = [ W λ C : L µ C ] + adj λµ + (cid:88) λ< J ν< J µ [ W λ C : L ν C ]adj νµ . The terms in the sum are all non-negative, so adj λµ = 0. (cid:3) James’s Conjecture.
Throughout the rest of this paper, we shall adopt the Kronecker delta.In view of the LLT algorithm (see Remark 2.16), the decomposition matrix D of S n or H n can becalculated in principle. Therefore, we focus our attention on studying the adjustment matrices A .The following is the famous James’s Conjecture for adjustment matrices. Conjecture 3.6. (James’s Conjecture [11, § ) Suppose that λ and µ are partitions lying in a blockB of S n or H n with e -weight w . If w < char( F ) , then adj λµ = δ λµ . James’s Conjecture is easy to verify for blocks of weight 0 or 1. A block of weight 0 containsits core κ as the only partition, so W κ = L κ regardless of the underlying field. Blocks of weight1 each contain e partitions which can be totally ordered by the dominance order, λ (cid:66) · · · (cid:66) λ e .The decomposition numbers are independent of the underlying field; [ W λ i : L λ j ] is equal to 1 if i ∈ { j, j + 1 } , and is equal to 0 otherwise. We summarize the progress on James’s Conjecture madeso far by the works of Fayers, Richards, Schroll and Tan in the following two theorems. heorem 3.7. (James’s Conjecture for blocks of Iwahori-Hecke algebras of weight at most 4) [8,Theorem 4.1] [9, Theorem 2.6] [19] Suppose that char( F ) ≥ . Let λ and µ be e -regular partitions lying in B , a block of H n of weightat most 4. Then, adj λµ = δ λµ . Theorem 3.8. (James’s Conjecture for blocks of q-Schur algebras of weight 2) [21, Corollary 3.6]
Suppose that char( F ) ≥ . Let λ and µ be partitions lying in B , a block of S n of weight 2. Then, adj λµ = δ λµ . In section 4, we prove James’s Conjecture for blocks of S n of weight 3. Theorem 3.9. (James’s Conjecture for blocks of q-Schur algebras of weight 3) Suppose that char( F ) ≥ . Let λ and µ be partitions lying in B , a block of S n of weight 3. Then, adj λµ = δ λµ . In section 5, we prove James’s Conjecture for blocks of S n of weight 4. Theorem 3.10. (James’s Conjecture for blocks of q-Schur algebras of weight 4) Suppose that char( F ) ≥ . Let λ and µ be partitions lying in B , a block of S n of weight 4. Then, adj λµ = δ λµ . In the context of Theorem 3.9 and Theorem 3.10, we already know that adj λµ = δ λµ when λ and µ are both e -regular due to Theorem 3.7. In fact, this knowledge can be strengthened for q -Schuralgebras: Proposition 3.11.
Let B be a block of S n and let ¯ B be its corresponding block of H n with the same e -core and e -weight. Let λ and µ be partitions lying in B with µ being e -regular. If the adjustmentmatrix for ¯ B is the identity matrix, then adj λµ = δ λµ .Proof. Suppose that λ (cid:54) = µ . Then,[ W λ F : L µ F ] = [ W λ C : L µ C ] + adj λµ + (cid:88) λ< J σ< J µ [ W λ C : L σ C ]adj σµ . Since µ is e -regular, we can apply Theorem 1.3 to get [ W λ F : L µ F ] = [ S λ F : D µ F ] and [ W λ C : L µ C ] =[ S λ C : D µ C ]. On the other hand, [ S λ F : D µ F ] = [ S λ C : D µ C ] since the adjustment matrix for ¯ B is theidentity matrix by assumption. Moreover, the terms in the sum are non-negative, so adj λµ mustbe zero. (cid:3) The row and column removal theorems.Definition 3.12.
We define the row removal function R as R : (cid:91) n> P ( n ) (cid:47) (cid:47) (cid:91) n> P ( n ) ν = ( ν , ν , . . . , ν l ( ν ) ) (cid:55)→ ( ν , ν , . . . , ν l ( ν ) ) . We define the column removal function C as C : (cid:91) n> P ( n ) (cid:47) (cid:47) (cid:91) n> P ( n ) ν = ( ν , ν , . . . , ν l ( ν ) ) (cid:55)→ ( ν − , ν − , . . . , ν l ( ν ) − . Theorem 3.13. ( [6, § ) Suppose that λ and µ are partitions of n with λ = µ . Then, [ W λ : L µ ] = [ W R ( λ ) : L R ( µ ) ] . Corollary 3.14. [16, Corollary 2.19]
Suppose that λ and µ are partitions of n with λ = µ . Then, adj λµ = adj R ( λ ) R ( µ ) . heorem 3.15. ( [6, § ) Suppose that λ and µ are partitions of n with l ( λ ) = l ( µ ) . Then, [ W λ : L µ ] = [ W C ( λ ) : L C ( µ ) ] . Corollary 3.16.
Suppose that λ and µ are partitions of n with l ( λ ) = l ( µ ) . Then, adj λµ = adj C ( λ ) C ( µ ) . Proof.
This is similar to the proof of Corollary 3.14 in [16, Corollary 2.19]; we use Theorem 3.15instead of Theorem 3.13. (cid:3)
Suppose that ν is a partition of weight w . Let us examine the weights of R ( ν ) and C ( ν ). Anabacus display for R ( ν ) is obtained from that of ν by replacing the bead corresponding to ν (themaximal occupied position) with an empty space. Therefore, if this bead has weight s ≥
0, then R ( ν ) would have weight w − s ≤ w . On the other hand, an abacus display for C ( ν ) is obtainedfrom that of ν by replacing the first unoccupied space with a bead. If there are r ≥ C ( ν ) would have weight w − r ≤ w . For our purposes, thecrucial point is that the weights of R ( ν ) and C ( ν ) are at most w . Remark . In sections 4 and 5, we will want to prove that adj λµ = δ λµ for all pairs of partitions( λ, µ ) lying in a block of S n of weight w , assuming that James’s conjecture holds for all blocks of S m of weight at most w , where m < n . Note that µ > P λ implies that µ ≥ λ and l ( µ ) ≤ l ( λ ). When µ = λ or l ( µ ) = l ( λ ), we may apply Corollary 3.14 or Corollary 3.16 respectively to conclude thatadj λµ = δ λµ . Thus, in this setting, we have adj λµ = 0 unless µ > P λ , µ > λ and l ( µ ) < l ( λ ). Wewrite µ (cid:29) λ when µ > P λ , µ > λ and l ( µ ) < l ( λ ).3.3. Lowerable partitions.
Recall the decomposition map d S between the Grothendieck groups G ( S n ) and G ( S n ). Suppose that A , B and C are blocks of S n − , S n and S n +1 respectively, and thatthere is an e -residue j such that a partition lying in A may be obtained from a partition lying in B by removing exactly one j -removable node, while a partition lying in C may be obtained from apartition lying in B by adding exactly one j -addable node. Let µ be an arbitrary partition in B .We define E j to be the j-restriction functor from G ( S n ) to G ( S n − ) and F j to be the j-induction functor from G ( S n ) to G ( S n +1 ) in the following way: E j ([ M ]) := [ M ↓ A ] ,F j ([ M ]) := [ M ↑ C ] . Similarly, we define ¯ E j to be the j-restriction functor from G ( S n ) to G ( S n − ) and ¯ F j to be the j-induction functor from G ( S n ) to G ( S n +1 ) in the following way:¯ E j ([ M ]) := [ M ↓ A ] , ¯ F j ([ M ]) := [ M ↑ C ] . It is easy to check using Theorem 2.18 that d S ¯ E j ([ W µ C ]) = E j d S ([ W µ C ]) ,d S ¯ F j ([ W µ C ]) = F j d S ([ W µ C ]) . Since { [ W µ C ] | µ ∈ P ( n ) } is a basis for G ( S n ), the following diagrams commute. G ( S n − ) G ( S n − ) d S (cid:47) (cid:47) G ( S n ) G ( S n − ) ¯ E j (cid:15) (cid:15) G ( S n ) G ( S n ) d S (cid:47) (cid:47) G ( S n ) G ( S n − ) E j (cid:15) (cid:15) G ( S n +1 ) G ( S n +1 ) d S (cid:47) (cid:47) G ( S n ) G ( S n +1 ) ¯ F j (cid:15) (cid:15) G ( S n ) G ( S n ) d S (cid:47) (cid:47) G ( S n ) G ( S n +1 ) F j (cid:15) (cid:15) ince the adjustment matrices are unitriangular, d S must be bijective; we shall identify G ( S n ) withits image under d S . Under this identification, we get ¯ E j = E j and ¯ F j = F j from the commutativediagrams. In particular, E j ([ L µ F ]) = [ L µ F ↓ A ] ,F j ([ L µ F ]) = [ L µ F ↑ C ] ,E j ([ L µ C ]) = [ L µ C ↓ A ] ,F j ([ L µ C ]) = [ L µ C ↑ C ] . Let t be a positive integer. We denote the divided power j-restriction functor and divided powerj-induction functor as E ( t ) j := t ! E tj and F ( t ) j := t ! F tj respectively. Proposition 3.18.
Let λ and µ be two distinct partitions lying in some block B of S n . If (cid:15) j ( λ ) <(cid:15) j ( µ ) (resp. ϕ j ( λ ) < ϕ j ( µ ) ) for some e -residue j , then adj λµ = 0 .If (cid:15) j ( λ ) = (cid:15) j ( µ ) (resp. ϕ j ( λ ) = ϕ j ( µ ) ), for some e -residue j , then adj λµ = adj ˜ E kj λ ˜ E kj µ (resp. adj λµ = adj ˜ F kj λ ˜ F kj µ ), where k := (cid:15) j ( λ ) (resp. k := ϕ j ( λ ) ).Proof. Let l := (cid:15) j ( µ ) and k := (cid:15) j ( λ ). If l > k , then E ( l ) j ([ L λ C ]) = (cid:88) ν ∈P ( n ) adj λν E ( l ) j [ L ν F ] . By Theorem 2.20, E ( l ) j ([ L λ C ]) = 0 and E ( l ) j [ L µ F ] (cid:54) = 0, so adj λµ must be zero.If l = k , then(3.1) [ L ˜ E kj λ C ] = E ( k ) j ([ L λ C ]) = (cid:88) ν ∈P ( n ) adj λν E ( k ) j [ L ν F ] = (cid:88) ν ∈P ( n ) ,(cid:15) j ( ν ) ≥ k adj λν E ( k ) j [ L ν F ] = (cid:88) ν ∈P ( n ) ,(cid:15) j ( ν )= k adj λν [ L ˜ E kj ν F ] , where the third equality is due to Theorem 2.20 and the last equality is due to the case l > k thatwe just proved above. On the other hand,(3.2) [ L ˜ E kj λ C ] = (cid:88) σ ∈P ( n − k ) adj ˜ E kj λ,σ [ L σ F ] . Comparing the coefficients of [ L ˜ E kj µ F ] in equations 3.1 and 3.2, we conclude that adj λµ = adj ˜ E kj λ ˜ E kj µ .The proof of the other case considering conormal nodes is similar. (cid:3) Remark . In sections 4 and 5, we will want to prove that adj λµ = δ λµ for all pairs of partitions( λ, µ ) lying in a block of S n of weight w , assuming that James’s conjecture holds for all blocks of S m of weight at most w , where m < n . If (cid:15) j ( µ ) > (cid:15) j ( λ ) ≤ (cid:15) j ( µ ) for some e -residue j , thenwe may apply Proposition 3.18 to conclude that adj λµ = δ λµ . Definition 3.20.
Let λ and µ be two distinct partitions lying in some weight w block B of S n .We say that the pair ( λ, µ ) is lowerable if there is some e -residue j such that (cid:15) j ( µ ) > ϕ j ( µ ) > (cid:15) j ( λ ) ≤ (cid:15) j ( µ ). Corollary 3.21.
Suppose that λ and µ are two distinct partitions lying in some weight w block B of S n and that ( λ, µ ) is lowerable. Moreover, suppose that the adjustment matrix is the identitymatrix for blocks of weight less than w . Then adj λµ = 0 .Proof. Let k := (cid:15) j ( µ ) and l := ϕ j ( µ ). If (cid:15) j ( λ ) < k , this follows directly from Proposition 3.18. If (cid:15) j ( λ ) = k , then we have adj λµ = adj ˜ E kj λ ˜ E kj µ by Proposition 3.18. By Lemma 2.21, ˜ E kj λ has weight w − kl < w , so the result follows. (cid:3) ote that our notion of lowerable partitions here is inspired by and generalises Fayers’s definitionof lowerable partitions in [9, Proposition 2.17]. Example 3.22.
Suppose that e = 8 and w = 5. Let λ = (cid:104) , , , , (cid:105) and µ = (cid:104) , , , , (cid:105) be thepartitions lying in the block B of H n with the (cid:104) e (cid:105) notation. λ µ Observe that (cid:15) ( µ ) = 1, ϕ ( µ ) = 1 and (cid:15) ( λ ) = 0. Hence, ( λ, µ ) is lowerable by Theorem 3.7.3.4. [ w:k ]- pairs . Let B be a weight w block of S n whose core κ B has exactly k removable nodeson a given runner i with residue j (if 0 < i < e , runner i has exactly k more beads than runner i −
1. If i = 0, runner 0 has exactly k + 1 more beads than runner e − A is theweight w block of S n − k whose core κ A is obtained from κ B by removing all of the k j -removablenodes. We say that the blocks A and B form a [ w : k ]-pair. We note that for each partition µ in A (resp. B ), we have ϕ j ( µ ) − (cid:15) j ( µ ) = k (resp. (cid:15) j ( µ ) − ϕ j ( µ ) = k ).Given a partition µ in A , recall that ˜ F kj µ is the partition in B obtained from µ by adding the k lowest j -conormal nodes. Then, ˜ F kj is a bijection from the set of partitions in A to the set ofpartitions in B . Moreover, we have the following: Theorem 3.23. [2]
Let µ be a partition in A . Then, ϕ j ( µ ) = k (equivalently (cid:15) j ( µ ) = 0 ) if andonly if (cid:15) j ( ˜ F kj µ ) = k , in which case, L µ ↑ B ∼ = ( L ˜ F kj µ ) ⊕ k ! and L ˜ F kj µ ↓ A ∼ = ( L µ ) ⊕ k ! . If this happens, wesay that µ and ˜ F kj µ are non-exceptional for the [ w : k ] -pair ( A, B ) . We say that µ and ˜ F kj µ areexceptional otherwise. When w ≤ k , every partition is non-exceptional for the [ w : k ]-pair ( A, B ), and we say that A and B are Scopes equivalent ; they are in fact Morita equivalent [20].
Remark . Let λ be a partition in A . • If λ were exceptional, then ϕ j ( λ ) > k (equivalently (cid:15) j ( λ ) > • If λ were non-exceptional, then ϕ j ( λ ) = k (equivalently (cid:15) j ( λ ) = 0).Let σ be a partition in B . • If σ were exceptional, then (cid:15) j ( σ ) > k (equivalently ϕ j ( σ ) > • If σ were non-exceptional, then (cid:15) j ( σ ) = k (equivalently ϕ j ( σ ) = 0). Definition 3.25.
Let µ ∈ P ( n ) and let j be some e -residue. If ϕ j ( µ ) − (cid:15) j ( µ ) = k >
0, we define˙ F j µ to be ˜ F kj µ ( ˙ F j µ is defined only when ϕ j ( µ ) − (cid:15) j ( µ ) > m , we define˙ F j (cid:38) m µ and ˙ F j (cid:37) m µ recursively in the following way: • If ϕ j ( µ ) − (cid:15) j ( µ ) >
0, ˙ F j (cid:38) µ := ˙ F j µ ( ˙ F j (cid:38) µ is not defined when ϕ j ( µ ) − (cid:15) j ( µ ) ≤ • If m > ϕ j − m +1 ( ˙ F j (cid:38) m − µ ) − (cid:15) j − m +1 ( ˙ F j (cid:38) m − µ ) >
0, ˙ F j (cid:38) m µ := ˙ F j − m +1 ( ˙ F j (cid:38) m − µ ). • If ϕ j ( µ ) − (cid:15) j ( µ ) >
0, ˙ F j (cid:37) µ := ˙ F j µ ( ˙ F j (cid:37) µ is not defined when ϕ j ( µ ) − (cid:15) j ( µ ) ≤ • If m > ϕ j + m − ( ˙ F j (cid:37) m − µ ) − (cid:15) j + m − ( ˙ F j (cid:37) m − µ ) >
0, ˙ F j (cid:37) m µ := ˙ F j + m − ( ˙ F j (cid:37) m − µ ). Proposition 3.26.
Suppose that A and B are blocks forming a [ w : k ] -pair as above with k < w .Let λ and µ be two distinct partitions in A .(1) If λ and µ are both non-exceptional, then adj λµ = adj ˙ F j λ, ˙ F j µ .(2) If λ is non-exceptional but µ is exceptional, then adj λµ = 0 .Proof. (1) By Remark 3.24, ϕ j ( λ ) = ϕ j ( µ ) = k , so adj λµ = adj ˙ F j λ, ˙ F j µ by Proposition 3.18.
2) By Remark 3.24, ϕ j ( λ ) = k and ϕ j ( µ ) > k , so adj λµ = 0 by Proposition 3.18. (cid:3) Suppose that we have a series of t + 1 blocks B m of weight w with cores κ m , where 0 ≤ m ≤ t . Moreover, for 1 ≤ m ≤ t , B m and B m − forms a [ w : k m ]-pair with κ m − being obtainedfrom κ m by adding k m > j m . Given any partition λ in B t , f ( λ ) :=˙ F j ˙ F j · · · ˙ F j t − ˙ F j t λ is well-defined. We say that f ( λ ) is semisimply induced if λ , ˙ F j t λ , ˙ F j t − ˙ F j t λ , . . . , ˙ F j · · · ˙ F j t − ˙ F j t λ and f ( λ ) are all non-exceptional. We say that f ( λ ) is not semisimply induced otherwise. Using this notation, we may restate our working version of Proposition 3.26 in thefollowing corollary. Corollary 3.27.
We adopt the notation above. Let λ and µ be two distinct partitions in B t .(1) If f ( λ ) and f ( µ ) are both semisimply induced, then adj λµ = adj f ( λ ) f ( µ ) .(2) If f ( λ ) is semisimply induced but f ( µ ) is not semisimply induced, then adj λµ = 0 .Proof. We just apply Proposition 3.26 repeatedly. (cid:3)
When we want to show that adj λµ = 0 for some pair of partitions ( λ, µ ) in section 5, we sometimesdo this by finding a sequence of e -residues j , . . . , j t such that f ( λ ) := ˙ F j · · · ˙ F j t λ is semisimplyinduced and ( f ( λ ) , f ( µ )) is lowerable. We may also use this formalism without making f explicitby writing λ ∼ ν (and say that λ induces semisimply to ν ) to indicate that there exists a sequenceof e -residues j , . . . , j t such that f ( λ ) := ˙ F j · · · ˙ F j t λ is semisimply induced and equals ν . Whenwe do so, it is hoped that it will not be too difficult for the reader to construct an appropriatesequence j , . . . , j t . Example 3.28.
Let λ = (cid:104) , (cid:105) , µ = (cid:104) , , (cid:105) and ν = (cid:104) , (cid:105) be the partitions lying in theweight 4 block B of S n with the (cid:104) , , (cid:105) notation (the beads on runner 0 have e -residue 5). Forany partition σ in B , define a ( σ ) := ˙ F (cid:37) σ = ˙ F ˙ F ˙ F σ . λ µ ν a ( λ ) a ( µ )Observe that a ( λ ) and a ( µ ) are semisimply induced and that ( a ( λ ) , a ( µ )) is lowerable. However, a ( ν ) is not semisimply induced.We state the following observation from the list of exceptional partitions of weights 3 and 4 inthe appendix (Figure 1, Figure 2 and Figure 3). Remark . Adopting the same notation as the beginning of this section, let A and B be blocksforming a [ w : k ]-pair with k < w and w ∈ { , } . Let λ be a partition in A . • If λ were exceptional, then ϕ j ( λ ) = k + 1 (equivalently (cid:15) j ( λ ) = 1). • If λ were non-exceptional, then ϕ j ( λ ) = k (equivalently (cid:15) j ( λ ) = 0).Let σ be a partition in B . If σ were exceptional, then (cid:15) j ( σ ) = k + 1 (equivalently ϕ j ( σ ) = 1). • If σ were non-exceptional, then (cid:15) j ( σ ) = k (equivalently ϕ j ( σ ) = 0).Therefore, we have the following result specific to blocks of S n of weights 3 and 4. Proposition 3.30.
Suppose that A and B are blocks forming a [ w : k ] -pair as above, with k < w and w ∈ { , } . Let λ and µ be two distinct partitions in A . If µ is exceptional, then both pairs ofpartitions ( λ, µ ) and ( ˙ F j λ, ˙ F j µ ) are lowerable.Proof. This follows directly from Remark 3.29 and Definition 3.20. (cid:3)
Rouquier blocks.
A weight w block B of S n with the (cid:104) b , . . . , b e − (cid:105) notation is Rouquier iffor every 0 ≤ i < j ≤ e −
1, either b i − b j ≥ w or b j − b i ≥ w −
1. The Rouquier blocks areScopes equivalent to each other and are now well understood. In particular, we know that James’sConjecture holds for Rouquier blocks.
Theorem 3.31. [12, Corollary 3.15] [3] If B is a Rouquier block of weight w < char( F ) , then adj λµ = δ λµ for all λ and µ in B . We say that a partition λ induces semi-simply to a Rouquier block if λ ∼ ν for some ν lying ina Rouquier block. As a consequence of Corollary 3.27 and Theorem 3.31, we have the followingresult. Proposition 3.32.
Suppose that λ and µ are partitions lying in a block B of S n of weight w < char( F ) . If λ induces semi-simply to a Rouquier block, then adj λµ = δ λµ . In sections 4 and 5, we will sometimes state a partition ν lying in a Rouquier block and invitethe reader to verify that λ ∼ ν for some partition λ for which we want to show that adj λµ = δ λµ .3.6. Outline of the proof of James’s Conjecture for the blocks of q -Schur algebras ofweights 3 and 4. From now on, we assume that char( F ) ≥ ≤ w ≤
4. We prove Theorem3.9 and Theorem 3.10 by induction on n , with the base case being the unique weight w block of S we . We will deal with the base case at the beginning of sections 4 and 5.For the inductive step, we use [ w : k ]-pairs. If B is a weight w block of S n and n > we , then thereis at least one block A forming a [ w : k ]-pair with B . Suppose that A , . . . , A t are all the blocks with A m forming a [ w : k m ]-pair with B , for each m . If λ is a partition in B which is non-exceptionalfor some pair ( A m , B ), then adj λµ = δ λµ for every partition µ lying in B by Proposition 3.26 andinduction.Therefore, we may assume that λ is exceptional for every pair ( A m , B ). Suppose that an abacusdisplay for the core of A m is obtained from that of B by removing k m removable nodes on runner i m , for each m . If λ were exceptional for ( A m , B ), then there must be at least k m + 1 normal beadson runner i m in the abacus display for λ . Hence, | λ ( i m ) | ≥ k m + 1 for each m (see Figure 1, Figure2 and Figure 3), so we have ( k + 1) + · · · + ( k t + 1) ≤ | λ ( i ) | + · · · + | λ ( i t ) | ≤ w . When w = 3, thisimplies that t = 1 and k ≤
2. When w = 4, this implies that either t = 1 and k ≤ t = 2 and k = k = 1. The blocks B satisfying these conditions are dealt with in the rest of the paper.By Remark 3.17, we may always assume that µ (cid:29) λ ; that is µ > P λ , µ > λ and l ( µ ) < l ( λ ).Additionally, for each e -residue j such that (cid:15) j ( µ ) >
0, we may assume that (cid:15) j ( λ ) < (cid:15) j ( µ ) by Remark3.19. Finally, we may assume that µ is e -singular by Proposition 3.11 and Theorem 3.7.4. Proof of James’s Conjecture for weight 3 blocks of S n In this section, we will first prove Theorem 3.9 and then use it to show that the decompositionnumbers for weight 3 blocks of S n are bounded above by one. Whenever λ and µ are partitionswith weight less than 3, adj λµ = δ λµ by Theorem 3.8. We will use this fact repeatedly withoutfurther comment. .1. The principal block of S e . Let B be the principal block of S e ; that is the weight 3 blockwhich we display on an abacus with the (cid:104) e (cid:105) notation. Proposition 4.1.
Suppose that λ and µ are partitions lying in B . Then, adj λµ = δ λµ .Proof. Let λ be an arbitrary partition lying in B . Observe that each runner of λ has at most onenormal bead. By Remark 3.19, we may assume that µ has no normal beads on any runner. ByRemark 3.17, we may also assume that µ (cid:29) λ . Table 1 (in the appendix) lists all the possiblepairs of partitions ( λ, µ ). We invite the reader to check that every partition in column λ of Table1 induces semi-simply to a Rouquier block. • (cid:104) (cid:105) ∼ (cid:104) | , , . . . , e + 1 (cid:105) , • (cid:104) (cid:105) ∼ (cid:104) , | , , . . . , e + 1 (cid:105) , • (cid:104) , (cid:105) ∼ (cid:104) , | , , . . . , e + 1 (cid:105) , • (cid:104) , (cid:105) ∼ (cid:104) , | , , . . . , e + 1 (cid:105) . Proposition 3.32 completes the proof of Proposition 4.1. (cid:3)
Blocks forming exactly one [3 : 1] -pair.
In this section, we prove the following proposition:
Proposition 4.2.
Suppose that A and B are weight blocks of S n − and S n respectively, forminga [3 : 1] -pair. Moreover, suppose that there is no block other than A forming a [3 : k ] -pair with B for any k . Additionally, suppose that the adjustment matrix for every weight block of S m is theidentity matrix whenever m < n . Then, the adjustment matrix for B is the identity matrix. The conditions on A and B mean that we may B on an abacus with the (cid:104) a , b − a , e − b (cid:105) notation,where 0 < a < b ≤ e . Suppose that λ and µ are two distinct partitions lying in B , so we wantto prove that adj λµ = 0. By Proposition 3.26 and Proposition 3.30, we may assume that λ isexceptional and that µ is non-exceptional. We list all the exceptional partitions in B below: • (cid:104) a , (cid:105) , • (cid:104) a , i (cid:105) , a < i < e , • (cid:104) i, a (cid:105) , 0 ≤ i ≤ a − • (cid:104) a (cid:105) .We invite the reader to check that (cid:104) a (cid:105) induces semi-simply to a Rouquier block, so we may assumethat λ (cid:54) = (cid:104) a (cid:105) by Proposition 3.32. (cid:104) a (cid:105) ∼ (cid:40) (cid:104) , a, a + e − b | , , . . . , e + 1 (cid:105) if e − b > , (cid:104) , a | , , . . . , e + 1 (cid:105) if e − b = 0 . (cid:104) , a (cid:105) (cid:104) a , (cid:105) Let ν := (cid:104) , a (cid:105) . Note that l ( ν ) ≥ l ( λ ) for all the remaining possible λ , so we may assume that l ( µ ) < l ( ν ) by Remark 3.17. Since the first unoccupied position in the abacus display for λ occurs atposition 0, the first unoccupied position in an abacus display for µ must occur strictly after position0. Consequently, all beads in runner 0 of an abacus display for µ have zero weight, therefore µ must be e -regular and adj λµ = δ λµ by Proposition 3.11 and Theorem 3.7. This completes the proofof Proposition 4.2. .3. Blocks forming exactly one [3 : 2] -pair.Proposition 4.3.
Suppose that A and B are weight blocks of S n − and S n respectively, forminga [3 : 2] -pair. Moreover, suppose that there is no block other than A forming a [3 : k ] -pair with B for any k . Additionally, suppose that the adjustment matrix for every weight block of S m is theidentity matrix whenever m < n . Then, the adjustment matrix for B is the identity matrix. The conditions on B mean that we may represent B on an abacus with the (cid:104) a , b − a , c − b , e − c (cid:105) notation, where 0 < a < b ≤ c ≤ e . If λ and µ are two distinct partitions in B , then we haveadj λµ = 0 by Proposition 3.26 unless λ is the unique exceptional partition for ( A, B ), namely λ = (cid:104) a (cid:105) . λ Observe that the first unoccupied position in the abacus display for λ occurs at position a . ByRemark 3.17, we may assume that l ( µ ) < l ( λ ), so the first unoccupied position in an abacus displayfor µ must occur after position a . Consequently, all beads in runner 0 of an abacus display for µ have zero weight, therefore µ must be e -regular and adj λµ = δ λµ by Proposition 3.11 and Theorem3.7.As discussed in section 3.6, the combination of Proposition 4.1, Proposition 4.2 and Proposition4.3 completes the proof of Theorem 3.9.In his weight 3 paper, Fayers proved an upper bound for the decomposition numbers of H n . Theorem 4.4. [8, Theorem 1.1]
Suppose that char( F ) ≥ and that B is a block of H n of weight3. Let λ and µ be partitions in B , with µ being e -regular. Then, [ S λ : D µ ] ≤ . An easy consequence of Theorem 3.9 is that we may extend this upper bound to the case of the q -Schur algebras. Corollary 4.5.
Suppose that char( F ) ≥ and that B is a block of S n of weight 3. Let λ and µ bepartitions in B . Then, [ W λ : L µ ] ≤ . Proof. If µ is e -regular, then [ W λ : L µ ] = [ S λ : D µ ] by Theorem 1.3, so we are done by Theorem4.4. If µ is e -singular, we display λ and µ on an abacus with e runners and r beads, for some r large enough. Then, we define two abacus displays with e + 1 runners each by adding a runnerwith every space unoccupied to the right of all the existing runners in the abacus displays for λ and µ (see example 2.17). Let λ + and µ + be the partitions corresponding to these two new abacusdisplays. Theorem 2.14 applies and so we have d eλµ ( v ) = d e +1 λ + µ + ( v ). Moreover, µ + is ( e + 1)-regular,so [ W λ C : L µ C ] = [ W λ + C : L µ + C ] = [ S λ + C : D µ + C ] ≤ λ + and µ + have weight 3). By Theorem 3.9, [ W λ C : L µ C ] = [ W λ F : L µ F ]. (cid:3) Proof of James’s Conjecture for weight 4 blocks of S n We shall prove Theorem 3.10 in this section. Whenever λ and µ are partitions with weight lessthan 4, adj λµ = δ λµ by Theorem 3.8 and Theorem 3.9. We will use this fact repeatedly withoutfurther comment. .1. The principal block of S e . Let B be the principal block of S e ; that is the weight 4 blockwhich we display on an abacus with the (cid:104) e (cid:105) notation. Proposition 5.1.
Suppose that λ and µ are partitions lying in B . Then, adj λµ = δ λµ .Proof. Observe that when λ = (cid:104) i (cid:105) and 0 < i < e , λ has two normal beads on runner i and nonormal beads on every other runner. By Remark 3.19, we may assume that µ has at most onenormal bead on runner i and no normal beads on every other runner. By Remark 3.17, we mayalso assume that µ (cid:29) λ . We list all the possiblities for µ below: • (cid:104) i (cid:105) , • (cid:104) i , i + 1 (cid:105) , i + 1 < e , • (cid:104) , i (cid:105) , • (cid:104) i , ( i + 1) (cid:105) , i + 1 < e .Notice that every partition µ in the list above are e -regular, therefore adj λµ = δ λµ by Proposition3.11 and Theorem 3.7.Hence, we may assume that λ (cid:54) = (cid:104) i (cid:105) for 0 < i < e . In this case, observe that in the abacusdisplays for the remaining possibilities for λ , no runner has more than one normal bead. By Remark3.19, we may assume that µ has no normal beads on any runner. By Remark 3.17, we may alsoassume that µ (cid:29) λ . Table 2 and Table 3 (in the appendix) list all the possible pairs of partitions( λ, µ ). Note that for each µ , we list the partitions λ in descending lexicographic order.We invite the reader to check that the following partitions induce semisimply to a Rouquierblock, thus adj λµ = δ λµ if λ is one of those partitions by Proposition 3.32. • (cid:104) , (cid:105) ∼ (cid:104) , | , (cid:105) when e = 2, • (cid:104) , , (cid:105) ∼ (cid:104) , | , (cid:105) when e = 2, • (cid:104) , , (cid:105) ∼ (cid:104) , | , , . . . , e + 1 (cid:105) , • (cid:104) , , (cid:105) ∼ (cid:104) , | , , . . . , e + 1 (cid:105) , • (cid:104) , , (cid:105) ∼ (cid:104) , | , , . . . , e + 1 (cid:105) , • (cid:104) , (cid:105) ∼ (cid:104) , | , , . . . , e + 1 (cid:105) , • (cid:104) , (cid:105) ∼ (cid:104) , | , , . . . , e + 1 (cid:105) , • (cid:104) , (cid:105) ∼ (cid:104) , , | , , . . . , e + 1 (cid:105) , • (cid:104) (cid:105) ∼ (cid:104) , | , , . . . , e + 1 (cid:105) , • (cid:104) , (cid:105) ∼ (cid:104) | , , . . . , e + 1 (cid:105) , • (cid:104) , (cid:105) ∼ (cid:104) , , | , , . . . , e + 1 (cid:105) , • (cid:104) , (cid:105) ∼ (cid:104) , | , , . . . , e + 1 (cid:105) , • (cid:104) (cid:105) ∼ (cid:104) , | , , . . . , e + 1 (cid:105) , • (cid:104) (cid:105) ∼ (cid:104) | , , . . . , e + 1 (cid:105) .Let λ , λ , λ , µ , µ and µ (see Table 2 and Table 3) be the following partitions: • λ := (cid:104) , (cid:105) , e = 3, • µ := (cid:104) , , (cid:105) , e = 3, • λ := (cid:104) , (cid:105) , e ≥ • µ := (cid:104) , , , (cid:105) , e ≥ • λ := (cid:104) (cid:105) , e = 2, • µ := (cid:104) , (cid:105) , e = 2.If we managed to show that adj λ µ = adj λ µ = adj λ µ = 0, the proof of Proposition 5.1 wouldfollow from Proposition 3.4.We now relax the definition of λ and µ slightly, so that λ := (cid:104) | , e − (cid:105) and µ := (cid:104) , | , e − (cid:105) , where e ≥
2. We also define λ := (cid:104) , | , e − (cid:105) , where e ≥
2. Recall from Definition3.25 that ˙ F (cid:38) e − ˙ F (cid:38) e − ν = ˙ F . . . ˙ F ˙ F ˙ F . . . ˙ F e − ˙ F ν is well-defined for partitions ν in the block B . When i ∈ { , } , we invite the reader to check (in an abacus display with the (cid:104) e (cid:105) notation, thebeads on runner 0 have e -residue 0) that ˙ F (cid:38) e − ˙ F (cid:38) e − λ i and ˙ F (cid:38) e − ˙ F (cid:38) e − µ i are semisimply nduced and moreover, ˙ F (cid:38) e − ˙ F (cid:38) e − λ i = λ and ˙ F (cid:38) e − ˙ F (cid:38) e − µ i = µ . Therefore, we are leftto show that adj λ µ = 0 and adj λ µ = 0 by Corollary 3.27. Proposition 5.2. adj λ µ = 0 .Proof. Let f ( λ ) := ˙ F (cid:38) e − λ and f ( µ ) := ˙ F (cid:38) e − µ (in an abacus display with the (cid:104) , e − (cid:105) notation, the beads on runner 0 have e -residue 2. When e = 2, f is the identity map). We observethat f ( λ ) and f ( µ ) are both semisimply induced and moreover, f ( λ ) = (cid:104) ( e − | e − , , (cid:105) and f ( µ ) = (cid:104) ( e − , ( e − | e − , , (cid:105) . Hence, adj λ µ = adj f ( λ ) f ( µ ) by Corollary 3.27. Wealso note that d ef ( λ ) f ( µ ) ( v ) = d λ µ ( v ) = v + v by Theorem 2.15. Hence, [ W f ( λ ) C : L f ( µ ) C ] = 2and it suffices to prove that [ W f ( λ ) F : L f ( µ ) F ] = 2 by Lemma 3.5. λ f ( λ ) µ f ( µ )Let B i be the weight 4 block with the (cid:104) e − i − , , i , (cid:105) notation for 0 ≤ i ≤ e −
2. We define λ y , λ x and µ x to be the partitions lying in B by their abacus displays below. We may check using themodular branching rules (Theorem 2.18 and Theorem 2.20) that W f ( λ ) F ↑ B ∼ W λ y F + W λ x F ,L f ( µ ) F ↑ B ∼ = L µ x F .λ y λ x µ x We define λ y,x , λ x,x and µ x,x to be the partitions lying in B e − by their abacus displays below.We may check using the modular branching rules that W λ y F ↑ B B ↑ B B ↑ · · · ↑ B e − B e − ∼ W λ y,x F ,W λ x F ↑ B B ↑ B B ↑ · · · ↑ B e − B e − ∼ W λ x,x F ,L µ x F ↑ B B ↑ B B ↑ · · · ↑ B e − B e − ∼ = L µ x,x F .λ y,x λ x,x µ x,x Let C be the weight 4 block with the (cid:104) , e − , (cid:105) notation. We define λ y,x,x , λ x,x,x , λ x,x,y , λ x,x,z and µ x,x,x to be the partitions lying in C by their abacus displays below. We may check using themodular branching rules that W λ y,x F ↑ CB e − ∼ ( W λ y,x,x F ) , λ x,x F ↑ CB e − ∼ ( W λ x,x,x F ) + ( W λ x,x,y F ) + ( W λ x,x,z F ) ,L µ x,x F ↑ CB e − ∼ = L µ x,x,x F ⊕ L µ x,x,x F . Hence, we have the upper bound2[ W f ( λ ) F : L f ( µ ) F ] = [ W f ( λ ) F : L f ( µ ) F ][ L f ( µ ) F ↑ B ↑ B B · · · ↑ B e − B e − ↑ CB e − : L µ x,x,x F ] ≤ (cid:88) ν [ W f ( λ ) F : L ν F ][ L ν F ↑ B ↑ B B · · · ↑ B e − B e − ↑ CB e − : L µ x,x,x F ]= [ W f ( λ ) F ↑ B ↑ B B · · · ↑ B e − B e − ↑ CB e − : L µ x,x,x F ]= 2([ W λ y,x,x F : L µ x,x,x F ] + [ W λ x,x,x F : L µ x,x,x F ] + [ W λ x,x,y F : L µ x,x,x F ] + [ W λ x,x,z F : L µ x,x,x F ]) .λ y,x,x λ x,x,x λ x,x,y λ x,x,z µ x,x,x Using the LLT algorithm and Theorem 2.15, we have d eλ y,x,x µ x,x,x ( v ) = v , d eλ x,x,x µ x,x,x ( v ) = v , d eλ x,x,y µ x,x,x ( v ) = 0 and d eλ x,x,z µ x,x,x ( v ) = 0. Additionally, observe that λ y,x,xi = λ x,x,xi = λ x,x,yi = µ x,x,xi for 1 ≤ i ≤ e −
1, so we may combine Corollary 3.14 (applied e − W λ y,x,x F : L µ x,x,x F ] = [ W λ y,x,x C : L µ x,x,x C ] = 1, [ W λ x,x,x F : L µ x,x,x F ] = [ W λ x,x,x C : L µ x,x,x C ] = 1and [ W λ x,x,y F : L µ x,x,x F ] = [ W λ x,x,y C : L µ x,x,x C ] = 0.By Theorem 3.1 and Corollary 3.3, we have[ W λ x,x,z F : L µ x,x,x F ] = [ W λ x,x,z C : L µ x,x,x C ] + adj λ x,x,z µ x,x,x + (cid:88) λ x,x,z < J ν< J µ x,x,x [ W λ x,x,z C : L ν C ]adj νµ x,x,x . From the abacus display of µ x,x,x , we observe that R e − ( µ x,x,x ) has weight 3. Therefore, theterms in the sum above are non-zero only if λ x,x,z < J ν < J µ x,x,x and ν e − < µ e − by Corollary3.14, Corollary 3.16 and Theorem 3.9. It is easy to check that the set { ν : λ x,x,z < J ν < J µ x,x,x and ν e − < µ e − } is empty. Moreover, λ x,x,z ∼ (cid:104) , | , , . . . , e + 1 (cid:105) induces semisimplyto a Rouquier block, so adj λ x,x,z µ x,x,x = 0. Additionally, [ W λ x,x,z C : L µ x,x,x C ] = d eλ x,x,z µ x,x,x (1) = 0,therefore [ W λ x,x,z F : L µ x,x,x F ] = 0 and [ W f ( λ ) F : L f ( µ ) F ] = 2. (cid:3) To prove Proposition 5.1, all that remains is to show that adj λ µ = 0. Since d eλ µ ( v ) = 0(calculated using the LLT algorithm and runner removal theorems 2.14 and 2.15), we just need tocheck that adj νµ = 0 for every partition ν satisfying λ < P ν < P µ in order to apply Proposition3.4.We list all the partitions ν such that λ < P ν < P µ : • ν = (cid:104) , | , e − (cid:105) ∼ (cid:104) , | , , . . . , e + 1 (cid:105) , so adj νµ = 0 by Proposition 3.32. • ν = (cid:104) , , | , e − (cid:105) ∼ (cid:104) , | , , . . . , e + 1 (cid:105) , so adj νµ = 0 by Proposition 3.32. • ν = λ , so adj λ µ = 0 by Proposition 5.2.This concludes the proof of Proposition 5.1. (cid:3) Blocks forming exactly two [4 : 1] -pairs.
In the next two sections, we prove the followingproposition. roposition 5.3. Let B be a weight block of S n . Suppose that there are exactly two blocks A and A forming a [4 : 1] -pair with B , and that there are no other blocks C forming a [4 : k ] -pairwith B for any k . Additionally, suppose that the adjustment matrix for every weight block of S m is the identity matrix whenever m < n . Then, the adjustment matrix for B is the identity matrix. The conditions above give two distinct types of block B .5.2.1. Blocks with the (cid:104) a , b − a , c − b , d − c , e − d (cid:105) notation. We first consider the case where B hasthe (cid:104) a , b − a , c − b , d − c , e − d (cid:105) notation, for some 0 < a < b < c < d ≤ e . Let λ and µ be twodistinct partitions in B for which we want to prove that adj λµ = 0. By Proposition 3.26, if λ werenon-exceptional for any one of ( A i , B ), then adj λµ = δ λµ . Therefore, we may assume that λ isexceptional for both pairs ( A , B ) and ( A , B ); that is λ = (cid:104) a , c (cid:105) . λ Observe that the first unoccupied position in the abacus display for λ above occurs at position a .By Remark 3.17, we may assume that l ( µ ) < l ( λ ), so the first unoccupied position in an abacusdisplay for µ must occur after position a . Consequently, all beads in runner 0 of an abacus displayfor µ have zero weight, therefore µ must be e -regular and adj λµ = δ λµ by Proposition 3.11 andTheorem 3.7.5.2.2. Blocks with the (cid:104) a , b − a , c − b , d − c , e − d (cid:105) notation. We complete the proof of Proposition5.3 by considering blocks B with the (cid:104) a , b − a , c − b , d − c , e − d (cid:105) notation, for some 0 < a < b < c ≤ d ≤ e . Let λ and µ be two distinct partitions in B for which we want to prove that adj λµ = 0. ByProposition 3.26, we may assume that λ is exceptional for both pairs ( A , B ) and ( A , B ); that is λ = (cid:104) a , b (cid:105) with b − a ≥ λ Observe that the first unoccupied position in the abacus display for λ above occurs at position a .By Remark 3.17, we may assume that l ( µ ) < l ( λ ), so the first unoccupied position in an abacusdisplay for µ must occur after position a . Consequently, all beads in runner 0 of an abacus displayfor µ have zero weight, therefore µ must be e -regular and adj λµ = δ λµ by Proposition 3.11 andTheorem 3.7.This completes the proof of Proposition 5.3.5.3. Blocks forming exactly one [4 : 3] -pair.Proposition 5.4.
Suppose that A and B are weight blocks of S n − and S n respectively, forminga [4 : 3] -pair. Moreover, suppose that there is no block other than A forming a [4 : k ] -pair with B for any k . Additionally, suppose that the adjustment matrix for every weight block of S m is theidentity matrix whenever m < n . Then, the adjustment matrix for B is the identity matrix.Proof. The conditions on B mean that we may represent B on an abacus with the (cid:104) a , b − a , c − b , d − c , e − d (cid:105) notation, where 0 < a < b ≤ c ≤ d ≤ e . If λ and µ are two distinct partitions in B , then we have dj λµ = 0 by Proposition 3.26 unless λ is the unique exceptional partition for ( A, B ), namely λ = (cid:104) a (cid:105) . λ Observe that the first unoccupied position in the abacus display for λ above occurs at position a . By Remark 3.17, we may assume that l ( µ ) < l ( λ ), so the first unoccupied position in an abacusdisplay for µ must occur after position a . Consequently, all beads in runner 0 of an abacus displayfor µ have zero weight, therefore µ must be e -regular and adj λµ = δ λµ by Proposition 3.11 andTheorem 3.7. (cid:3) Blocks forming exactly one [4 : 2] -pair.Proposition 5.5.
Suppose that A and B are weight blocks of S n − and S n respectively, forminga [4 : 2] -pair. Moreover, suppose that there is no block other than A forming a [4 : k ] -pair with B for any k . Additionally, suppose that the adjustment matrix for every weight block of S m is theidentity matrix whenever m < n . Then, the adjustment matrix for B is the identity matrix. The conditions on B mean that we may represent B on an abacus with the (cid:104) a , b − a , c − b , e − c (cid:105) notation, where 0 < a < b ≤ c ≤ e . Suppose that λ and µ are two distinct partitions lying in B and we want to prove that adj λµ = 0. If λ were non-exceptional for ( A, B ), then adj λµ = δ λµ byProposition 3.26. Therefore we may assume that λ is exceptional; that is λ must be one of thefollowing partitions: • (cid:104) a , (cid:105) , • (cid:104) a , i (cid:105) , i / ∈ { a − , a } , • (cid:104) a (cid:105) .We invite the reader to verify that (cid:104) a (cid:105) induces semi-simply to a Rouquier block: (cid:104) a (cid:105) ∼ (cid:104) , a, a + e − c, a + e − b | , , . . . , e + 1 (cid:105) if e − c > , c − b > , (cid:104) , a , a + e − c | , , . . . , e + 1 (cid:105) if e − c = 0 , c − b > , (cid:104) , a, ( a + e − c ) | , , . . . , e + 1 (cid:105) if e − c > , c − b = 0 , (cid:104) , a | , , . . . , e + 1 (cid:105) if e − c = 0 , c − b = 0 . Therefore, we may assume that λ (cid:54) = (cid:104) a (cid:105) by Proposition 3.32. We are left to consider the cases λ = (cid:104) a , (cid:105) or λ = (cid:104) a , i (cid:105) , i / ∈ { a − , a } . (cid:104) , a (cid:105) (cid:104) a , (cid:105) Let ν := (cid:104) , a (cid:105) . Note that l ( ν ) ≥ l ( λ ) for all the remaining possible λ , so we may assume that l ( µ ) < l ( ν ) by Remark 3.17. Since the first unoccupied position in the abacus display for λ occurs atposition 0, the first unoccupied position in an abacus display for µ must occur strictly after position0. Consequently, all beads in runner 0 of an abacus display for µ have zero weight, therefore µ must be e -regular and adj λµ = δ λµ by Proposition 3.11 and Theorem 3.7. This completes the proofof Proposition 5.5. .5. Blocks forming exactly one [4 : 1] -pair.
In this section, we prove the following proposition:
Proposition 5.6.
Suppose that A and B are weight blocks of S n − and S n respectively, forminga [4 : 1] -pair. Moreover, suppose that there is no block other than A forming a [4 : k ] -pair with B for any k . Additionally, suppose that the adjustment matrix for every weight block of S m is theidentity matrix whenever m < n . Then, the adjustment matrix for B is the identity matrix. The conditions on B mean that we may represent B on an abacus with the (cid:104) a , b − a , e − b (cid:105) notation, where 0 < a < b ≤ e . Suppose that λ and µ are two distinct partitions lying in B and wewant to prove that adj λµ = 0. By Proposition 3.26 and Proposition 3.30 we may assume that λ isexceptional, so that λ is one of the following partitions: • (cid:104) a , (cid:105) , • (cid:104) a (cid:105) , • (cid:104) a , (cid:105) , • (cid:104) a , , i (cid:105) , i / ∈ { a − , a } , e ≥ • (cid:104) a , i (cid:105) , i / ∈ { a − , a } , e ≥ • (cid:104) a , i, j (cid:105) , i, j / ∈ { a − , a } and i < j , e ≥ • (cid:104) a , i (cid:105) , i (cid:54) = a , • (cid:104) a , i (cid:105) , i / ∈ { a − , a } , e ≥ • (cid:104) a (cid:105) .We invite the reader to check that (cid:104) a (cid:105) induces semi-simply to a Rouquier block: (cid:104) a (cid:105) ∼ (cid:40) (cid:104) , a, a + e − b | , , . . . , e + 1 (cid:105) if e − b > , (cid:104) , a | , , . . . , e + 1 (cid:105) if e − b = 0 . Therefore, we may assume that λ (cid:54) = (cid:104) a (cid:105) by Proposition 3.32. In the remaining possibilities for λ ,observe that λ has exactly two normal beads on runner a and at most one normal bead on each ofthe other runners. By Remark 3.19, we may assume that µ has at most one normal bead on runner a (non-exceptional) and no normal beads on every other runner. Note that ν := (cid:104) , a (cid:105) ≤ P λ for all the remaining possibilities for λ . By Remark 3.17 we may further assume that µ (cid:29) ν since µ (cid:29) λ and λ ≥ P ν imply that µ (cid:29) ν . By Corollary 3.11, we may also assume that µ is e -singular.We list every possibility for µ satisfying all the conditions stated above:(A1) (cid:104) a , b (cid:105) , e − b > (cid:104) a, b , (cid:105) , b − a = 1 , e − b > (cid:104) a, a + 1 , b (cid:105) , b − a ≥ , e − b > (cid:104) , a (cid:105) , a = 1 , b − a = 1 , e − b = 0,(B2) (cid:104) , a, a + 1 (cid:105) , a = 1 , e − b = 0, b − a ≥ (cid:104) , a (cid:105) ,(C2) (cid:104) , a , a + 1 (cid:105) , b − a ≥ (cid:104) , a , (cid:105) , a = 1,(C4) (cid:104) , , a (cid:105) , a ≥ (cid:104) , a, a + 1 , a + 2 (cid:105) , b − a ≥ (cid:104) , a , a + 1 (cid:105) , a = 1 , b − a ≥ (cid:104) , , a, a + 1 (cid:105) , a ≥ , b − a ≥ The case e − b > . For any partition ν lying in B , we define a ( ν ) := ˙ F a − b (cid:37) a ν (in an abacusdisplay with the (cid:104) a , b − a , e − b (cid:105) notation, the beads on runner 0 have e -residue a − b ). We maycheck that a ( λ ) and a ( µ ) are both semisimply induced and that ( a ( λ ) , a ( µ )) is lowerable exceptwhen one of the following happens: • λ = (cid:104) a (cid:105) or λ = (cid:104) i , a (cid:105) for some 0 ≤ i ≤ a − e − b = 1 and µ is in case A2 or A3.When λ ∈ {(cid:104) a (cid:105) , (cid:104) i , a (cid:105)} , none of our ten cases of µ (cases A1-A3, C1-C7) satisfy µ (cid:29) λ , thereforeadj λµ = δ λµ by Remark 3.17. In view of Corollary 3.21 and Corollary 3.27, we are left to consider µ in cases A2 and A3 where e − b = 1. By considering µ (cid:29) λ again, we are left with the followingpossibilities for λ : • (cid:104) a , i (cid:105) , 0 ≤ i ≤ a − i = b , • (cid:104) i , a (cid:105) , 0 ≤ i ≤ a − ν lying in B , we define b ( ν ) := ˙ F a (cid:38) b − a ν (in an abacus display with the (cid:104) a , b − a , e − b (cid:105) notation, the beads on runner b have e -residue a ). We observe that for the re-maining pairs ( λ, µ ), b ( λ ) and b ( µ ) are both semisimply induced and that ( b ( λ ) , b ( µ )) is lowerable,so the proof of Proposition 5.6 when e − b > The case e − b = 0 . The case e = b is much more difficult to deal with. From now on, weassume that e = b . We will first refine the list of possible µ . We note that (cid:104) , a (cid:105) ∼ (cid:104) , a | , , . . . , e + 1 (cid:105) , so we may assume that λ (cid:54) = (cid:104) , a (cid:105) by Proposition 3.32. In view of this, we have τ := (cid:104) a , a + 1 (cid:105) ≤ P λ for the remaining possibilities for λ when a = 1 and e − a ≥
2. When a = 1and e − a = 1, we have γ := (cid:104) a , (cid:105) ≤ P λ for the remaining possibilities for λ .By Remark 3.17, we may assume that µ (cid:29) τ when a = 1 and e − a ≥
2, since µ (cid:29) λ and λ ≥ P τ imply that µ (cid:29) τ ; when a = 1 and e − a = 1, we may assume that µ (cid:29) γ . These two additionalrestrictions produce the following refined list of possibilities for µ :(C1) (cid:104) , a (cid:105) ,(C2) (cid:104) , a , a + 1 (cid:105) , e − a ≥ (cid:104) , a , (cid:105) , a = 1, e − a ≥ (cid:104) , , a (cid:105) , a ≥ (cid:104) , a, a + 1 , a + 2 (cid:105) , e − a ≥ (cid:104) , , a, a + 1 (cid:105) , a ≥ , e − a ≥ σ in B , define a ( σ ) := ˙ F a (cid:37) a σ (in an abacus display with the (cid:104) a , e − a (cid:105) notation,the beads on runner 0 have e -residue a ). When µ is in cases C1, C3 and C4 with e − a ≥ a ( λ ) and a ( µ ) are both semisimply induced and ( a ( λ ) , a ( µ )) is lowerable. When e − a ≥ µ is in caseC2, a ( λ ) and a ( µ ) are both semisimply induced and ( a ( λ ) , a ( µ )) is lowerable. In view of Corollary3.21 and Corollary 3.27, we are left with the following possibilities for µ :(C1) (cid:104) , a (cid:105) , e − a = 1,(C2) (cid:104) , a , a + 1 (cid:105) , e − a = 2,(C4) (cid:104) , , a (cid:105) , a ≥ e − a = 1,(C5) (cid:104) , a, a + 1 , a + 2 (cid:105) , e − a ≥ (cid:104) , , a, a + 1 (cid:105) , a ≥ , e − a ≥ Cases C1, C2 and C5 . Suppose that µ is in either case C1, C2 or C5. When e − a = 1, (cid:104) a , (cid:105) ∼ (cid:104) , a , | , , . . . , e + 1 (cid:105) , so we may assume that λ (cid:54) = (cid:104) a , (cid:105) by Proposition 3.32. ByRemark 3.17, we may also assume that λ (cid:28) µ . We find that the only remaining possibilities for λ are: • (cid:104) i , a (cid:105) for some 0 ≤ i ≤ a − • (cid:104) i, a (cid:105) for some 1 ≤ i ≤ a − λ, µ ) is to first induce them up semisimply viathe same sequence of inductions (for example using f := ˙ F a +2 (cid:38) e − a − . . . ˙ F a +1 (cid:38) e − a − ˙ F a (cid:38) e − a − ˙ F a − (cid:38) e − a . . . ˙ F a +1 (cid:38) e − a ˙ F a (cid:38) e − a which is well-definedfor partitions in block B ) to the block with the (cid:104) a , , e − a − (cid:105) notation, followed by an application ofthe Jantzen-Schaper formula. We denote their induced counterparts as (ˆ λ, ˆ µ ); that is ( λ, µ ) ∼ (ˆ λ, ˆ µ ). sing Theorem 2.6, we would show that [ W ˆ λ : L ˆ µ ] is independent of char( F ), so that adj ˆ λ ˆ µ = δ ˆ λ ˆ µ and hence adj λµ = δ λµ by Corollary 3.27 and Lemma 3.5. We may check that • (cid:104) i , a (cid:105) ∼ (cid:104) i , , a | a , , e − a − (cid:105) when i ≤ a − • (cid:104) i, a (cid:105) ∼ (cid:104) , i , a | a , , e − a − (cid:105) when 1 ≤ i ≤ a − • µ ∼ (cid:104) , a | a , , e − a − (cid:105) .Note that (cid:104) , , a | a , , e − a − (cid:105) ≤ J ˆ λ for all remaining possible λ . Table 4, Table 5 and Table6 (in the appendix) illustrates how we use Theorem 2.6. The entries of the tables are the Jantzen-Schaper coefficients (Definition 2.4) J ( ν, σ ), for partitions (cid:104) , , a | a , , e − a − (cid:105) ≤ J ν < J σ ≤ J ˆ µ = (cid:104) , a | a , , e − a − (cid:105) . Note that J F ( ν, σ ) = J C ( ν, σ ) since char( F ) > w = 4. We also omitthe columns indexed by ν if [ W ν : L ˆ µ ] = 0 as these do not contribute to our calculations. If weare able to justify the last column of Table 4, Table 5 and Table 6, it would imply that [ W ν : L ˆ µ ]is independent of F , hence adj ν ˆ µ = δ ν ˆ µ for all (cid:104) , , a | a , , e − a − (cid:105) ≤ J ν ≤ J ˆ µ ; this shows thatadj λµ = δ λµ for every remaining possible λ by Corollary 3.27 and Lemma 3.5.We now proceed to justify the last column of Table 4, Table 5 and Table 6. When B ( ν, ˆ µ ) ≤ W ν : L ˆ µ ] = B ( ν, ˆ µ ) by Corollary 2.7, so adj ν ˆ µ = δ ν ˆ µ by Lemma 3.5. When B ( ν, ˆ µ ) > ν := (cid:104) ( a − , , a | a , , e − a − (cid:105) . Observe that B ( ν , ˆ µ ) = 2 and that this is the onlypartition ν in Table 4, Table 5 and Table 6 with B ( ν, ˆ µ ) > σ e ( ν ) and σ e (ˆ µ ), and find that they are both ( − a , hence [ W ν C : L ˆ µ C ] = 1. Apriori, we only know that [ W ν F : L ˆ µ F ] ≤ Lemma 5.7.
We have [ W ν F : L ˆ µ F ] = 1 .Proof. We may check that ν induces semi-simply to a Rouquier block: ν ∼ (cid:40) (cid:104) , , a | , , . . . , e + 1 (cid:105) if a = 1 , (cid:104) , ( a − , a | , , . . . , e + 1 (cid:105) if a ≥ . Hence, adj ν ˆ µ = 0 by Proposition 3.32. Moreover, we deduce from the rows above ν in Table 4,Table 5 and Table 6 that [ W ν : L ˆ µ ] is independent of F whenever ν < J ν < J ˆ µ , therefore adj ν ˆ µ = 0by Lemma 3.5. Finally,[ W ν F : L ˆ µ F ] = (cid:88) ν ≤ J ν ≤ J ˆ µ [ W ν C : L ν C ]adj ν ˆ µ = [ W ν C : L ˆ µ C ] = 1 . (cid:3) Cases C4 and C7 . Suppose that µ is in either case C4 or C7. When e − a = 1, (cid:104) a , (cid:105) ∼ (cid:104) , a , | , , . . . , e + 1 (cid:105) , so we may assume that λ (cid:54) = (cid:104) a , (cid:105) by Proposition 3.32. By Remark 3.17, we mayalso assume that λ (cid:28) µ . We find that the only remaining possibilities for λ are: • (cid:104) , a (cid:105) , • (cid:104) , a (cid:105) , • (cid:104) , a (cid:105) .We use the Jantzen Schaper formula and Corollary 3.27 in the same fashion as in section5.5.2 to deal with the remaining pairs of ( λ, µ ). We may check that (for example using f :=˙ F a +2 (cid:38) e − a − . . . ˙ F a +1 (cid:38) e − a − ˙ F a (cid:38) e − a − ˙ F a − (cid:38) e − a . . . ˙ F a +1 (cid:38) e − a ˙ F a (cid:38) e − a ) • (cid:104) , a (cid:105) ∼ (cid:104) , , a | a , , e − a − (cid:105) , • (cid:104) , a (cid:105) ∼ (cid:104) , , a | a , , e − a − (cid:105) , • (cid:104) , a (cid:105) ∼ (cid:104) , , a | a , , e − a − (cid:105) , • µ ∼ (cid:104) , , a | a , , e − a − (cid:105) . ote that (cid:104) , , a | a , , e − a − (cid:105) < J (cid:104) , , a | a , , e − a − (cid:105) < J (cid:104) , , a | a , , e − a − (cid:105) . Theentries of Table 7 (in the appendix) are the Jantzen-Schaper coefficients (Definition 2.4) J ( ν, σ ),for partitions (cid:104) , , a | a , , e − a − (cid:105) ≤ J ν < J σ ≤ J ˆ µ := (cid:104) , , a | a , , e − a − (cid:105) . Note that J F ( ν, σ ) = J C ( ν, σ ) since char( F ) > w = 4. We also omit the columns indexed by ν if [ W ν : L ˆ µ ] = 0as these do not contribute to our calculations. If we are able to justify the last column of Table 7, itwould imply that [ W ν : L ˆ µ ] is independent of F , hence adj ν ˆ µ = δ ν ˆ µ for all (cid:104) , , a | a , , e − a − (cid:105) ≤ J ν ≤ J ˆ µ by Lemma 3.5; this shows that adj λµ = δ λµ for every remaining possible λ by Corollary3.27.We now proceed to justify the last column of Table 7. When B ( ν, ˆ µ ) ≤
1, we have [ W ν : L ˆ µ ] = B ( ν, ˆ µ ) by Corollary 2.7, so adj ν ˆ µ = δ ν ˆ µ by Lemma 3.5. When B ( ν, ˆ µ ) >
1, we have to do morework.Let ν := (cid:104) , , a | a , , e − a − (cid:105) . Observe that B ( ν , ˆ µ ) = 2 (shaded in Table 7) and that thisis the only partition ν in Table 7 with B ( ν, ˆ µ ) >
1. Following Remark 2.13, we calculate σ e ( ν )and σ e (ˆ µ ), and find that they are both +1, hence [ W ν C : L ˆ µ C ] = 1. A priori, we only know that[ W ν F : L ˆ µ F ] ≤ Lemma 5.8.
We have [ W ν F : L ˆ µ F ] = 1 .Proof. We may check that ν ∼ ˜ ν := (cid:104) , , | , , a − , e − a − (cid:105) and ˆ µ ∼ ˜ µ := (cid:104) , | , , a − , e − a − (cid:105) (see Figure 4 in the appendix). Observe that l (˜ ν ) = l (˜ µ ) and that C (˜ ν ) and C (˜ µ ) have weight 3. Hence adj ν ˆ µ = 0 by Theorem 3.9 and Corollary 3.16. Moreover,we deducefrom the rows above ν in Table 7 that [ W ν : L ˆ µ ] is independent of F whenever ν < J ν < J ˆ µ ,therefore adj ν ˆ µ = 0 by Lemma 3.5. Finally,[ W ν F : L ˆ µ F ] = (cid:88) ν ≤ J ν ≤ J ˆ µ [ W ν C : L ν C ]adj ν ˆ µ = [ W ν C : L ˆ µ C ] = 1 . (cid:3) This completes the proof of Proposition 5.6.As discussed in section 3.6, the combination of Proposition 5.1, Proposition 5.6, Proposition 5.5,Proposition 5.4 and Proposition 5.3 completes the proof of Theorem 3.10.
Acknowledgements . This paper was written under the supervision of Kai Meng Tan at theNational University of Singapore. The author would like to thank Prof Tan for his many helpfulcomments and guidance. The author is also grateful for the financial support given by the AcademicResearch Fund R-146-000-317-114 of NUS.
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J. Amer. Math. Soc. (2017). Appendix A. Figures and tablesFigure 1
Exceptional partitions for [3 : 1]-pairs.
Figure 2
Exceptional partitions for [3 : 2]-pairs. igure 3 Exceptional partitions for [4 : k ]-pairs, 0 < k < Table 1
Some cases ( λ, µ ) in section 4.1 µ λ (cid:104) (cid:105) NIL (cid:104) , e − (cid:105) , e = 2 (cid:104) (cid:105)(cid:104) (cid:105)(cid:104) , (cid:105)(cid:104) , (cid:105)(cid:104) , (cid:105) (cid:104) (cid:105)(cid:104) , , (cid:105) , e ≥ (cid:104) (cid:105)(cid:104) (cid:105)(cid:104) , (cid:105)(cid:104) , (cid:105) Department of Mathematics, National University of Singapore
Email address : [email protected] igure 4 ν = (cid:104) , , a | a , , e − a − (cid:105) , ˆ µ = (cid:104) , , a | a , , e − a − (cid:105) ν ∼ ∼ ˜ ν ∼ ∼ ˆ µ ∼ ∼ ˜ µ ∼ ∼ able 2 Some cases ( λ, µ ) in section 5.1. For each µ , the possiblities for λ are listedin descending lexicographic order µ λ d eλµ ( v ) (cid:104) (cid:105) NIL (cid:104) , (cid:105) (cid:104) (cid:105)(cid:104) , (cid:105) (cid:104) (cid:105)(cid:104) , , (cid:105) , e ≥ (cid:104) , (cid:105)(cid:104) , (cid:105)(cid:104) (cid:105)(cid:104) (cid:105)(cid:104) , , (cid:105) , e = 2 (cid:104) , (cid:105)(cid:104) , (cid:105)(cid:104) (cid:105)(cid:104) (cid:105) µ := (cid:104) , , (cid:105) , e = 3 (cid:104) , , (cid:105)(cid:104) , , (cid:105)(cid:104) , , (cid:105)(cid:104) , (cid:105)(cid:104) , (cid:105) v (cid:104) , (cid:105) v (cid:104) , (cid:105)(cid:104) , (cid:105) λ := (cid:104) , (cid:105) v (cid:104) (cid:105)(cid:104) , (cid:105)(cid:104) , (cid:105)(cid:104) , (cid:105)(cid:104) (cid:105)(cid:104) (cid:105) µ := (cid:104) , , , (cid:105) , e ≥ (cid:104) , , (cid:105)(cid:104) , , (cid:105)(cid:104) , , (cid:105)(cid:104) , (cid:105)(cid:104) , (cid:105) d λ ˆ µ ( v ) = v (cid:104) , (cid:105) d λ ˆ µ ( v ) = v (cid:104) , (cid:105)(cid:104) , (cid:105) λ := (cid:104) , (cid:105) d λ ˆ µ ( v ) = v (cid:104) (cid:105)(cid:104) , (cid:105)(cid:104) , (cid:105)(cid:104) , (cid:105)(cid:104) (cid:105)(cid:104) (cid:105) able 3 Some cases ( λ, µ ) in section 5.1. For each µ , the possibilities for λ arelisted in descending lexicographic order µ λ d eλµ ( v ) µ := (cid:104) , (cid:105) , e = 2 (cid:104) , (cid:105)(cid:104) , , (cid:105) λ := (cid:104) (cid:105) v + v (cid:104) , (cid:105) (cid:104) , (cid:105)(cid:104) , (cid:105)(cid:104) , (cid:105)(cid:104) (cid:105)(cid:104) (cid:105) Table 4
We have a > ν and ˆ µ = (cid:104) , a (cid:105) lie in the block with the (cid:104) a , , e − a − (cid:105) notation. See section 5.5.2 for an explanation of how we obtained the last column. (cid:104) , a (cid:105) (cid:104) , a (cid:105) (cid:104) ( a − , a (cid:105) (cid:104) , ( a − , a (cid:105) (cid:104) ( a − , , a (cid:105) (cid:104) ( a − , , a (cid:105) B ( ν, ˆ µ ) [ W ν : L ˆ µ ] (cid:104) , a (cid:105) (cid:104) , a , (cid:105) (cid:104) , a (cid:105) (cid:104) (cid:105) − (cid:104) , a (cid:105) − (cid:104) , a (cid:105) (cid:104) , a (cid:105) (cid:104) a (cid:105) (cid:104) , a − , a (cid:105) − (cid:104) , , a (cid:105) ( − a − ( − a (cid:104) ( a − , a (cid:105) (cid:104) ( a − , a (cid:105) − (cid:104) , a (cid:105) ( − a ( − a − (cid:104) , a (cid:105) ( − a − − a (cid:104) , a − , a (cid:105) (cid:104) , , a (cid:105) (cid:104) ( a − , a (cid:105) (cid:104) , a (cid:105) (cid:104) , a (cid:105) (cid:104) , ( a − , a (cid:105) (cid:104) , ( a − , a (cid:105) − (cid:104) , , a (cid:105) ( − a ( − a − (cid:104) ( a − , , a (cid:105) (cid:104) ( a − , , a (cid:105) (cid:104) ( a − , , a (cid:105) − (cid:104) , , a (cid:105) − a − ( − a (cid:104) , , a (cid:105) ( − a ( − a − able 5 We have a = 3, ν and ˆ µ = (cid:104) , (cid:105) lie in the block with the (cid:104) a , , e − a − (cid:105) notation. See section 5.5.2 for an explanation of how we obtained the last column. (cid:104) , (cid:105) (cid:104) , (cid:105) (cid:104) , (cid:105) (cid:104) , , (cid:105) (cid:104) , , (cid:105) (cid:104) , , a (cid:105) B ( ν, ˆ µ ) [ W ν : L ˆ µ ] (cid:104) , a (cid:105) (cid:104) , a , (cid:105) (cid:104) , (cid:105) (cid:104) (cid:105) − (cid:104) , a (cid:105) − (cid:104) , a (cid:105) (cid:104) , a (cid:105) (cid:104) a (cid:105) (cid:104) , , (cid:105) − (cid:104) , , (cid:105) − (cid:104) , a (cid:105) (cid:104) , a (cid:105) − (cid:104) , a (cid:105) − (cid:104) , , (cid:105) (cid:104) , , (cid:105) (cid:104) , (cid:105) (cid:104) , (cid:105) (cid:104) , (cid:105) (cid:104) , , (cid:105) (cid:104) , , (cid:105) − (cid:104) , , (cid:105) (cid:104) , , a (cid:105) (cid:104) , , a (cid:105) − Table 6
We have a = 2, ν and ˆ µ = (cid:104) , (cid:105) lie in the block with the (cid:104) a , , e − a − (cid:105) notation. See section 5.5.2 for an explanation of how we obtained the last column. (cid:104) , (cid:105) (cid:104) , (cid:105) (cid:104) , (cid:105) (cid:104) , , (cid:105) (cid:104) , , (cid:105) B ( ν, ˆ µ ) [ W ν : L ˆ µ ] (cid:104) , (cid:105) (cid:104) , , (cid:105) (cid:104) , (cid:105) (cid:104) (cid:105) − (cid:104) , (cid:105) − (cid:104) , (cid:105) (cid:104) , (cid:105) (cid:104) (cid:105) (cid:104) , , (cid:105) − (cid:104) , (cid:105) (cid:104) , (cid:105) − (cid:104) , , (cid:105) (cid:104) , (cid:105) (cid:104) , (cid:105) (cid:104) , , (cid:105) (cid:104) , , (cid:105) (cid:104) , , (cid:105) able 7 We have a ≥ e − a ≥ ν and ˆ µ = (cid:104) , , a (cid:105) lie in the block with the (cid:104) a , , e − a − (cid:105) notation. See section 5.5.2 for an explanation of how we obtained thelast column. (cid:104) , , a (cid:105) (cid:104) , , a (cid:105) (cid:104) , a (cid:105) (cid:104) , , a (cid:105) B ( ν, ˆ µ ) [ W ν : L ˆ µ ] (cid:104) , , a (cid:105) (cid:104) , , a (cid:105) (cid:104) , , a (cid:105) (cid:104) , a (cid:105) (cid:104) , a (cid:105) (cid:104) , , a (cid:105) (cid:104) , , a (cid:105) − (cid:104) , a (cid:105) − (cid:104) , , a (cid:105)1 1 1