On the modular McKay graph of SL_n(p) with respect to its standard representation
aa r X i v : . [ m a t h . R T ] F e b ON THE MODULAR MCKAY GRAPH OF SL n ( p ) WITH RESPECT TO ITSSTANDARD REPRESENTATION
MIRIAM G. NORRIS
Abstract
Let F be an algebraically closed field of prime characteristic p . The modular McKaygraph of G := SL n ( p ) with respect to its standard F G -module W is the connected, directedgraph whose vertices are the irreducible F G -modules and for which there is an edge from avertex V to V if V occurs as a composition factor of the tensor product V ⊗ W . We showthat the diameter of this modular McKay graph is ( p − n − n ). INTRODUCTION
Let G be a finite group, F an algebraically closed field. For an F G -module W the McKay graph M F ( G, W ) is the directed graph on the set of simple
F G -modules where there is an edge from V to V if V is a composition factor of V ⊗ W. Such graphs famously come up in the McKaycorrespondence which says for G a finite subgroup of SU ( C ) , F = C and W the standard 2-dimensional F G -module, M F ( G, W ) is an affine Dynkin diagram of type
A, D or E. We say thata McKay graph M F ( G, W ) is modular if the characteristic of F divides the order of | G | . It is a result of Burnside and Brauer [2] that an
F G -module W is faithful if and only if everyirreducible F G -module occurs as a composition factor of a tensor power of W. This implies thatfor any two irreducible
F G -modules V and V there exists some integer j such that V ⊗ V ∗ hasa composition factor in common with W ⊗ j . From this we can deduce that there exists is a pathfrom V to V of length j. This means when W is a faithful F G -module the graph M F ( G, W ) isconnected and it makes sense to look for results about its diameter.Benkart et. al. [1] initiated a study of mixing times for particular random walks on modularMcKay graphs. This built on work of Fulman [5] who considered random walks on McKay graphswhere F = C . A statement about the diameter of modular McKay graphs provides a lower boundfor the mixing times of the random walks considered in [1].Liebeck, Shalev and Tiep [7] observed that when G is a finite group and W is a faithful F G -module there is an obvious lower bound for the diameter: diam M F ( G, W ) ≥ log( b ( G ))log(dim W )1here b ( G ) is the largest dimension of an irreducible F G -module. For F = C , G a finite simplegroup and W an irreducible F G -module it is conjectured in [7] that this bound is tight up toa constant and this is proved when G has bounded rank. Motivated by this we consider themodular McKay graphs for a family of groups of unbounded rank. In this paper we prove thefollowing theorem concerning the graph Γ = M F ( G, V n ) where F is a field of characteristic p , G = SL n ( p ) and V n is the standard n -dimensional F G -module.
Theorem 1.1.
The diameter of the modular McKay graph of SL n ( p ) with respect to its standardmodule V n is ( p − n − n ) . Note that the irreducible
F G -module of largest dimension is the Steinberg module. Thishas dimension p ( n − n ) giving us a lower bound of ( n − n ) log( p )2 log( n ) which is much smaller than ( p − n − n ) suggesting very different behaviour from the cases considered in [7].We prove Theorem 1.1 in Section 4 by showing ( p − n − n ) is both a lower and an upperbound for the diameter of Γ. We find a lower bound using basic results about C SL n ( C )-modulesreviewed in Section 2. In Section 3 we use a result of Brundan and Kleshchev [Theorem V(iv),[3]]to prove the existence of certain edges in Γ . This allows us to show the lower bound is also anupper bound in Section 4.
Acknowledgements
The author gratefully acknowledges Martin Liebeck for suggesting the problem, guiding itsprogress and patiently reviewing preliminary versions of this work. The author would alsolike to acknowledge Alexander Kleshchev for an extremely helpful conversation. This work wassupported by the Engineering and Physical Sciences Research Council [EP/L015234/1] throughthe EPSRC Centre for Doctoral Training in Geometry and Number Theory (London School ofGeometry and Number Theory). THE MCKAY GRAPH OF SL n ( C )Let Γ C := M C ( SL n ( C ) , V n ) be the McKay graph of SL n ( C ) with respect to the standard n -dimensional C G -module V n . In this section we will review some basic facts about the ordinaryrepresentation theory of SL n ( C ) . This will enable us in Lemma 2.3 to obtain a restriction on theedges of Γ C .We begin with some notation. Let H = SL n ( C ) and denote by Φ the root system of type A n − corresponding to some maximal torus T ≤ H. Fix ∆ = { α , . . . , α n − } to be a base ofsimple roots in Φ and let E be the Euclidean space spanned by ∆. Let ∆ ∨ denote the set of simplecoroots and { λ , . . . , λ n − } the dual basis of fundamental dominant weights relative to the innerproduct on E. The Cartan matrix C expresses the change of basis from the set of fundamentalweights to ∆. A weight is a vector in E with integral inner product with the coroots and can bewritten as a integral combination of the fundamental dominant weights. Therefore if λ is a weight2e can write λ = P i m i λ i and hence think of it as a vector of integers m = ( m , . . . , m n − ); thisis the notation we will adopt throughout the paper. Applying the change of basis matrix C − allows us to write such a weight λ as a sum of simple roots, λ = P i c i α i such that C − m T = c T where c = ( c , . . . , c n − ) . We call a weight λ = P i m i λ i dominant if m i ≥ ≤ i ≤ n − λ ≤ µ and say λ is subdominantto µ if µ − λ has nonnegative coefficients when written as a sum of simple roots.The irreducible C H -modules are characterised by the set of dominant weights [see for example §
15 in [8]]. For a dominant weight λ we write its corresponding irreducible C H -module W C ( λ ) . These are the vertices of the graph Γ C . The irreducible C H -module corresponding to the n − , , . . . ,
0) is the n -dimensional standard module which we denoted by V n . Furthermorethe trivial module corresponds to the n − , . . . ,
0) which we will denote throughoutas 0 . For two dominant weights λ, µ we write λ → C µ if there is a directed edge in Γ C from W C ( λ ) to W C ( µ ). The composition factors of W C ( λ ) ⊗ V n are known as an application of theLittlewood-Richardson rule [see for example § . Lemma 2.1.
Let H = SL n ( C ) and let W C ( λ ) and V n be C H -modules as defined above with λ = ( m , . . . , m n − ) . Then W C ( µ ) is a composition factor of W C ( λ ) ⊗ V n if and only if µ takesthe form of one of the following:(a) ( m + 1 , m , . . . , m n − ) (b) ( m , . . . , m i − , m i − , m i +1 + 1 , . . . m n − ) for ≤ i ≤ n − . (c) ( m , . . . , m n − , m n − − . We now define a value associated to a dominant weight λ that will allow us to determine inLemma 2.3 a restriction on edges in Γ C . Definition 2.2.
Let λ be any dominant weight and let c n − be the coefficient of the simple root α n − in λ written as a sum of simple roots. We define the following integer associated to λ,f ( λ ) = nc n − . Clearly we have f (0) = 0 , furthermore if we let St p denote the weight ( p − , . . . , p −
1) then f ( St p ) = ( p − n − n ) . Let d ( λ, µ ) denote the distance from W C ( λ ) to W C ( µ ) in the graphΓ C . Lemma 2.3.
In the graph Γ C the following hold.1) If λ → C µ then f ( µ ) ≤ f ( λ ) + 1 .
2) Furthermore d (0 , St p ) = ( p − n − n ) . Proof.
As in the statement of Lemma 2.1 let λ = ( m , . . . , m n − ) and assume µ takes the form inoptions ( a ) , ( b ) or ( c ) . Inspecting the bottom row of the Cartan matrix of type A n − we see that3he coefficient of α n − in λ written as a sum of roots is P n − i =1 in m i . Therefore f ( µ ) = f ( λ ) + 1if µ takes the form ( a ) or ( b ) and f ( µ ) = f ( λ ) − ( n −
1) if µ takes the form ( c ) . In particular f ( µ ) ≤ f ( λ ) + 1 completing the proof of 1).If follows from 1) that d (0 , St p ) ≥ f ( St p ) = ( p − n − n ) hence to show 2) it will suffice tofind a path of length ( p − n − n ) from W C (0) to W C ( St p ) . We start at the vertex W C (0) andcall this stage 0 . At each stage j > W C ( λ ) where λ has p − j − a ) in Lemma 2.1 and then we repeatedly move along the edges described by option( b ) for i = 1 to i = n − j . We repeat this step p − p − n − j entry. Terminating this process after n − p − n = 5 , p = 2. W C (0 , , , stage 1 −−−−→ W C (1 , , , → W C (0 , , , → W C (0 , , , → W C (0 , , , stage 2 −−−−→ W C (1 , , , → W C (0 , , , → W C (0 , , , stage 3 −−−−→ W C (1 , , , → W C (0 , , , stage 4 −−−−→ W C (1 , , , . For each 0 < j ≤ n − j th stage requires passing along ( n − j )( p −
1) edges and thereforethe length of the path is ( p − n − n ) as required. EDGES IN THE MODULAR MCKAY GRAPH OF SL n ( p )In this section we will review some facts and introduce some notation that will allow us to provein Lemma 3.2 the existence of some edges in the modular McKay graph of SL n ( p ) . Let K = F p and denote by SL n the algebraic group SL n ( K ) . Let G = SL n ( p ) the fixedpoint group under the standard Frobenius map on SL n that sends ( a i,j ) ( a pi,j ) . Recall fromthe introduction that Γ = M K ( G, V n ) is the modular McKay graph of G = SL n ( p ) with respectto the standard n -dimensional KG -module V n . We briefly describe the relationship between C SL n ( C )-, KSL n - and KG -modules in the following discussion.As previously let W C ( λ ) be the irreducible C H -module indexed by a dominant weight λ . There exists a Z -form W Z ( λ ) contained in W C ( λ ) such that SL n acts on W ( λ ) = W Z ( λ ) ⊗ K [see for example § . KSL n -module W ( λ ) the Weyl module corresponding to λ . For each dominant weight λ the Weyl module W ( λ ) has a unique maximal submodule M ( λ )such that the quotient V ( λ ) = W ( λ ) /M ( λ ) is an irreducible KSL n -module. The compositionfactors of M ( λ ) are irreducible KSL n -modules V ( µ ) such that µ < λ under the dominancerelation described in the Section 2. The irreducible KG -modules are the restrictions of V ( λ )for p -restricted weights λ . For simplicity we will abuse notation and refer to V n as the C H -4odule W C (1 , , . . . , , the corresponding Weyl module W (1 , , . . . ,
0) and also its restriction to G. Note that W (1 , , . . . ,
0) = V (1 , , . . . ,
0) as there are no dominant weights subdominant to(1 , , . . . , . The vertices of Γ = M K ( G, V n ) are therefore parametrised by the set of p -restricted dominantweights of length n − . For two p -restricted weights λ, µ we write λ → µ if there is an edge in Γfrom V ( λ ) | G to V ( µ ) | G i.e. if V ( µ ) | G is a composition factor of V ( λ ) | G ⊗ V n . Let us now make some definitions that are necessary to state Theorem B (iv) from [3] whichwe then use to prove Lemma 3.2. To a dominant weight λ we can associate a partition ˜ λ oflength n such that λ i = ˜ λ i − ˜ λ i − and ˜ λ n = 0 . Let ǫ i be the n -tuple whose entries are all 0 exceptfor a 1 in position i .We say i is ˜ λ - addable if ˜ λ + ǫ i is a partition, that is ˜ λ i +1 ≤ ˜ λ i + 1 ≤ ˜ λ i − . We denote byAdd(˜ λ ) the set of ˜ λ -addable i. Similarly we say i is ˜ λ - removable if ˜ λ − ǫ i is a partition and denoteby Rem(˜ λ ) the set of ˜ λ -removable i. If i is ˜ λ -addable then we say it is conormal for ˜ λ if there is an injection g from the set R i = { k ∈ Rem(˜ λ ) : 1 ≤ k < i, ˜ λ i + 1 − i ≡ ˜ λ k − k mod p } into the set A i = { k ′ ∈ Add(˜ λ ) : 1 ≤ k ′ < i, ˜ λ i + 1 − i ≡ ˜ λ k ′ + 1 − k ′ mod p } such that g ( k ) > k for all k ∈ R i . Partitions ˜ µ of length n parametrise the irreducible KGL n ( K )-modules which we denote by V ′ (˜ µ ). Furthermore GL n ( K ) is the central product of SL n and K × where the action of K × on V ′ (˜ µ ) is the ˜ µ n th power of the determinant. Restricting KGL n ( K )-modules to the subgroup SL n we may ignore the action K × which is indicated in the partition by the n th entry. For apartition ˜ µ of length n we have V ′ (˜ µ , ˜ µ , . . . , ˜ µ n ) | SL n = V ( µ ′ ) where the partition correspondingto µ ′ is (˜ µ − ˜ µ n , ˜ µ − ˜ µ n , . . . , ˜ µ n − − ˜ µ n ) . In particular we note V ′ (1 , , . . . , | SL n = V n . Thefollowing theorem is a re-statement of Theorem B(iv) in [3].
Theorem 3.1.
For a partition ˜ µ of length n, let V ′ (˜ µ ) be the corresponding irreducible KGL n ( K ) -module. If i is conormal for ˜ µ then V ′ (˜ µ + ǫ i ) occurs as a composition factor of V ′ (˜ µ ) ⊗ V ′ (1 , , . . . , . The following result describes some edges in Γ . Lemma 3.2. If λ and µ are p -restricted weights satisfying one of the following three conditions,then λ → µ. λ is any p -restricted weight and µ is the unique element of the set { , . . . , p − } such that µ ≡ λ +1 mod p − , and µ i = λ i for i = 1 . ) λ is a p -restricted weight whose first nonzero entry is λ s for some ≤ s < n − , µ s = λ s − and µ s +1 is the unique element of the set { , . . . , p − } such that µ s +1 ≡ λ s +1 +1 mod p − , and µ i = λ i for i = s, s + 1 . λ is a p -restricted weight whose first nonzero entry is λ n − , µ n − = λ n − − and µ i = λ i = 0 for i = n − . Proof.
To show λ → µ it is enough to show that there is some weight µ ′ such that V ( µ ′ ) is acomposition factor of V ( λ ) ⊗ V n and V ( µ ) | G is a composition factor of V ( µ ′ ) | G . By Theorem 3.1if i is conormal for ˜ λ then V ′ (˜ λ + ǫ i ) | SL n is a composition factor of V ′ (˜ λ ) | SL n ⊗ V n . Further-more as discussed above V ′ (˜ λ ) | SL n ⊗ V ′ (1 , , . . . , SL n = V ( λ ) ⊗ V n . Therefore we will look forcomposition factors of V ′ (˜ λ + ǫ i ) | SL n for i conormal for ˜ λ. We begin by defining two possible values of i that are always conormal for ˜ λ. Groupingconsecutive entries of ˜ λ with the same values we can write ˜ λ in block form. That is, we write˜ λ = (˜ λ a , ˜ λ a a , . . . ) where the first a entries all have value ˜ λ and the next a entries havevalue ˜ λ a = ˜ λ and so on. We claim that 1 and 1 + a are always conormal for ˜ λ. To seethis for 1 is trivial since there can be no j ∈ Rem (˜ λ ) such that j < . To see that 1 + a isconormal note that the first a entries have the same value, so a is the only element of Rem (˜ λ )strictly smaller than 1 + a . However ˜ λ a and ˜ λ a are strictly less than p and ˜ λ a = ˜ λ a , so λ a λ a +1 mod p. Therefore the condition for 1 + a to be conormal is met.We now show that if 1) holds then λ → µ . By the remarks above V ′ (˜ λ + ǫ ) | SL n isa composition factor of V ( λ ) ⊗ V n . Since ˜ λ n = 0 there exists some weight µ ′ such that V ′ (˜ λ + ǫ ) | SL n = V ( µ ′ ) . In particular µ ′ = λ +1 and µ ′ i = λ i for i = 1 . Since λ is p -restricted µ ′ is p -restricted unless λ = p − . If µ ′ is p -restricted then µ ′ = µ and we have λ → µ. If λ = p − µ ′ = p and by Steinburg’s tensor product theorem we can write V ( µ ′ ) = V (0 , µ ′ , . . . , µ ′ n − ) ⊗ V (1 , , . . . , ( p ) . Restricting to G this gives us the KG -module( V (0 , µ ′ , . . . , µ ′ n − ) ⊗ V (1 , , . . . , | G . Furthermore we note µ = (1 , µ ′ , . . . , µ ′ n − ) and therefore V ( µ ) | G is a composition factor of ( V (0 , µ ′ , . . . , µ ′ n − ) ⊗ V (1 , , . . . , | G and hence λ → µ. Now suppose 2) holds. Since ˜ λ i = ˜ λ i +1 for 1 ≤ i ≤ a but ˜ λ a = ˜ λ a +1 we have that s = a . Therefore a + 1 < n and hence there exists some dominant weight µ ′ such that V ( µ ′ ) = V ′ (˜ λ + ǫ a +1 ) | SL n . In particular µ ′ a = λ a − µ ′ a +1 = λ a +1 +1 and λ i = µ i for i = a , a + 1 . Applying the same arguments as above we see that if λ s < p − µ ′ is p -restricted and µ = µ ′ and therefore λ → µ . Otherwise if λ s = p − µ = ( µ ′ , . . . , µ ′ s − , , µ ′ s +2 , . . . , µ ′ n − ) andtherefore V ( µ ) | G is a composition factor of V ( µ ′ ) | G implying λ → µ. Finally we suppose 3) holds. As remarked above if s = n − λ then a + 1 = n. Therefore ˜ λ + ǫ a +1 = (˜ λ , ˜ λ , . . . , ˜ λ n − , . Taking µ ′ to bethe weight such that V ( µ ′ ) = V ′ (˜ λ + ǫ a +1 ) | SL n we have ˜ µ ′ = (˜ λ − , ˜ λ − , . . . , ˜ λ n − − , . Therefore µ ′ n − = λ n − − µ ′ i = λ i = 0 for 1 ≤ i < n − µ = µ ′ . Therefore wehave λ → µ completing the proof. 6 emark . In the proofs in the next section we will refer to moving along an edge from V ( λ ) | G to V ( µ ) | G where λ and µ are as in part 1) of Lemma 3.2, as adding 1 to the first entry of λ . Likewise for λ and µ as in part 2) we will refer to this move as clearing a 1 from the first nonzeroentry of λ and adding it to the next entry. We will also refer to this process as moving 1 along.Finally for λ and µ as in part 3) of Lemma 3.2 we will refer to this move as clearing 1 from theentry in position n − λ . THE DIAMETER OF
ΓIn this section we will combine results from Section 2 and Section 3 to prove Theorem 1.1. Recallthat G = SL n ( p ) , K is an algebraically closed field of characteristic p and Theorem 1.1 concernsthe McKay graph Γ = M K ( G, V n ) where V n is the standard KG -module. Theorem 1.1.
The diameter of the modular McKay graph of SL n ( p ) with respect to its standardmodule V n is ( p − n − n ) . The proof of Theorem 1.1 follows immediately from Lemma 4.2 in which we show ( p − n − n ) is a lower bound for the diameter of Γ and Lemma 4.3 in which we show this is alsoan upper bound. For a weight λ recall the integer f ( λ ) from Definition 2.2 in Section 2. In orderto prove Lemma 4.2 we will need the following result. Proposition 4.1.
Let µ be a p -restricted weight and µ ′ a dominant weight for SL n ( K ) . If V ( µ ) | G is a composition factor of V ( µ ′ ) | G then f ( µ ) ≤ f ( µ ′ ) . Proof.
For a weight λ = ( λ , . . . , λ n − ) let S ( λ ) = P n − i =1 λ i . Note that if λ and ν are weights suchthat λ − ν written as a sum of roots corresponds to the n − c , . . . , c n − ) then applyingthe Cartan matrix of type A n − we see S ( λ − ν ) = c + c n − . Therefore if ν is subdominantto λ then S ( ν ) ≤ S ( λ ). We will prove Proposition 4.1 by induction on S ( µ ′ ). Let µ, µ ′ beas in the statement of the Proposition and observe that if µ ′ is p -restricted then µ = µ ′ andhence f ( µ ) = f ( µ ′ ) . We will now prove the inductive step. Suppose for all weights γ suchthat S ( γ ) < S ( µ ′ ) the proposition holds. By Steinberg’s tensor product theorem we may write V ( µ ′ ) = N i V ( ν i ) ( p i − ) and therefore V ( µ ′ ) | G = N i V ( ν i ) | G . Since we assume that V ( µ ) | G is a composition factor of V ( µ ′ ) | G = N i V ( ν i ) | G there exists a dominant weight γ such that V ( γ ) is a composition factor of N i V ( ν i ) and V ( µ ) | G is a composition factor of V ( γ ) | G . Notethat γ ≤ P i ν i and therefore f ( γ ) ≤ f ( P i ν i ) ≤ f ( µ ′ ) and S ( γ ) ≤ S ( P i ν i ). Since µ ′ isnot p -restricted we see S ( P i ν i ) = P i S ( ν i ) < S ( µ ′ ). Therefore by our inductive assumption f ( µ ) ≤ f ( γ ). Since f ( γ ) ≤ f ( µ ′ ) this completes the inductive step.For two p -restricted dominant weights, λ and µ let d Γ ( λ, µ ) denote the the distance from V ( λ ) | G to V ( µ ) | G in Γ . Furthermore let St p denote the weight ( p − , . . . , p −
1) of SL n ( K ) . Lemma 4.2.
In the graph Γ we have d Γ (0 , St p ) = ( p − n − n ) . Hence this is a lower boundon the diameter of Γ . roof. As in the proof of Lemma 2.3 we will first show that if λ and µ are two p -restricted weightssuch that λ → µ then f ( µ ) ≤ f ( λ ) + 1. This provides the desired lower bound on d (0 , St p ) since f (0) = 0 and f ( St p ) = ( p − n − n ) . Since λ → µ there exists a dominant weight µ ′ suchthat V ( µ ′ ) is a composition factor of V ( λ ) ⊗ V n and V ( µ ) | G is a composition factor of V ( µ ′ ) | G . By Proposition 4.1, f ( µ ) ≤ f ( µ ′ ) and so it remains to show that f ( µ ′ ) ≤ f ( λ ) + 1. Observe thatthe KSL n -module W ( λ ) ⊗ V n must contain the composition factors of V ( λ ) ⊗ V n , one of whichis V ( µ ′ ). Therefore µ ′ is subdominant to some weight η where the composition factors of theWeyl module W ( η ) are composition factors of W ( λ ) ⊗ V n . For such an η we have λ → C η andhence by Lemma 2.3, f ( η ) ≤ f ( λ ) + 1 . Finally since µ ′ ≤ η we have f ( µ ′ ) ≤ f ( λ ) + 1 . It now remains to show that a path of length ( p − n − n ) from V (0) | G to V ( St p ) | G exists. Recall the path described in the proof of Lemma 2.3. Note that any two consecutivevertices in that path are irreducible C H -modules whose highest weights λ and µ are p -restricted.Furthermore each such λ and µ satisfy one of the first two conditions in Lemma 3.2. Therefore V ( λ ) | G and V ( µ ) | G are vertices in Γ and by Lemma 3.2, λ → µ. This describes a path in Γ from V (0) | G to V ( St p ) | G of length ( p − n − n ).We now complete the proof of Theorem 1.1 by showing that ( p − n − n ) is also an upperbound for the diameter of Γ . For convenience we will refer to the vertices of Γ by the p -restricteddominant weights that parametrise them. Lemma 4.3.
For any two p -restricted weights λ, µ we have d Γ ( λ, µ ) ≤ ( p − n − n ) . Hencethis is an upper bound for the diameter of Γ .Proof. Let λ and µ be distinct p -restricted weights. We will show there is a path from V ( λ ) | G to V ( µ ) | G of length no more than ( p − n − n ) . In order to describe such a path we make thefollowing definitions. Let M be the set of vertices in the path detailed in the proof of Lemma4.2 from V (0) | G to V ( St p ) | G . For any p -restricted weight µ we define the following integer ℓ µ := µ = Stn µ = 0max { x ∈ Z | ≤ x ≤ n − , µ x < p − } otherwise . If µ ∈ M we say µ ∈ f M if all entries before position ℓ µ are 0 . We define the integers µ := µ ∈ f M max { x ∈ Z | ≤ x < ℓ µ , µ x > } otherwise . To a p -restricted weight µ we associate an element of M ( µ ) ∈ M defined as follows. If µ ∈ M let M ( µ ) = µ , otherwise let M ( µ ) := (0 , . . . , , , , . . . , , µ ℓ µ , p − , . . . , p −
1) where the 1 is inposition s µ and the µ ℓ µ is in position ℓ µ . By Lemma 3.2 there exists a path from M ( µ ) to µ as8ollows. If µ ∈ M then the path is trivial so we may assume 1 ≤ s µ < ℓ µ . We add 1 to the firstentry, move it along repeatedly until we add 1 in position s µ and repeat this µ s µ − µ is µ s µ . We then add 1 to the first entry, moveit along until it is in position s µ − µ s µ − times. This results in a weight in whichthe entry in position s µ − µ s µ − . Continue in this way until the resulting weight is µ. Thetotal number of steps taken in this path is( µ s µ −
1) s µ + s µ − X i =1 iµ i . We will now describe a path from λ to M ( µ ) for cases (i) ℓ λ > ℓ µ and (ii) ℓ λ ≤ ℓ µ . The pathexists by Lemma 3.2 where we are using the terminology from Remark 3.3.Suppose λ = 0 and λ and µ are as in (i). There is a path as follows. Starting at λ add 1 tothe first entry λ times where λ is the unique element of the set { , . . . , p − } such that λ + ℓ λ X i =1 λ i ≡ p − . The resulting weight is one in which the entry in the first position is 0 if λ = λ = 0 and theunique integer S ∈ { , , . . . , p − } such that S ≡ λ + λ mod p − S times resulting in a weight in which the first entry is 0 . Continue in thisway; for 1 < j ≤ ℓ λ let j − j − j repeatedly until the resulting weight has 0 in position j −
1. The entry inthe position j of this weight will be 0 if P ji =0 λ i = 0 and the unique integer S j ∈ { , , . . . , p − } such that S j ≡ P ji =0 λ i mod p − λ thatif P ℓ λ i =0 λ i = 0 then in the final step clearing 1’s from position ℓ λ − ℓ λ is p −
1. Therefore we have described a path from λ to a weightin which all entries before position ℓ λ are 0 and the entry in position ℓ λ is 0 if P ℓ λ i =0 λ i = 0 and p − ℓ λ in the resulting weight will be p − . Thispath takes λ + P ℓ λ − i =1 S i steps. Then add 1 to the first entry and move it along to position ℓ λ − p − p − ℓ µ are p − . We then add 1 and move italong to position ℓ µ and repeat µ ℓ µ times. Finally add 1 and move it to position s µ . This pathtakes a further P ℓ λ − i = ℓ µ +1 i ( p −
1) + ℓ µ µ ℓ µ + s µ steps and the resulting weight is M ( µ ) . Letting X denote the number of edges in the path above from λ to M ( µ ) and then to µ we9ave X = λ + ℓ λ − X i =1 S i + ℓ λ − X i = ℓ µ +1 i ( p −
1) + ℓ µ µ ℓ µ + s µ X i =1 iµ i = λ + ℓ λ − X i =1 S i + ℓ λ − X i = ℓ µ +1 i ( p −
1) + ℓ µ X i =1 iµ i . Note that λ and each of the S i are all no greater than p − ℓ λ < n which implies λ + P ℓ λ − i =1 S i ≤ ( n − p − . Furthermore since each µ i ≤ p − X ≤ P n − i =1 ( p − i = ( p − n − n ) . Suppose now that λ = 0 . The path from λ to M ( µ ) is the path described in Lemma 4.2 since M ( µ ) ∈ M. If µ ∈ M then d Γ ( λ, µ ) ≤ ( p − n − n ) by Lemma 4.2. Therefore we may assume µ M in which case, looking at the definition of M ( µ ) , we see that the path from 0 to M ( µ )described by Lemma 4.2 is of length X ′ = s µ + ℓ µ µ ℓ µ + n − − ℓ µ X i =1 ( n − i )( p − . Therefore if X is the number of edges in the path from 0 to µ then X = X ′ + ( µ s µ −
1) s µ + s µ − X i =1 iµ i ≤ n − − ℓ µ X i =1 ( n − i )( p −
1) + ℓ µ µ ℓ µ + s µ X i =1 iµ i ≤
12 ( p − n − n ) . Suppose from now on that λ and µ are as in case (ii) so that ℓ λ ≤ ℓ µ . Note that ℓ µ = 0 ifand only if ℓ λ = 0 and therefore µ = St p . If µ ℓ µ = 0 then there is a path from λ to µ as follows. Starting with λ add 1 to the first entry λ times where λ is the unique element of the set { , . . . , p − } such that λ + ℓ µ X i =1 λ i ≡ µ ℓ µ mod p − . Then clear 1’s repeatedly from the first nonzero entry until the entry in every position before ℓ µ in the resulting weight is 0. By Lemma 3.2 this describes a path to a weight in which all entriesbefore position ℓ µ are 0 and the entry in position ℓ µ is S ℓ µ = µ ℓ µ , where we adopt the notation S i from the proof of case (i). This process takes λ + P ℓ µ − i =1 S i steps. Then add a 1 and move italong to position s µ . This describes a path from λ to the weight M ( µ ) . The resulting path from10 to M ( µ ) and then to µ therefore takes X steps where X = λ + ℓ µ − X i =1 S i + ℓ µ − X i =1 iµ i ≤
12 ( p − n − n ) . If µ ℓ µ = 0 and ℓ µ = n − λ to M ( µ ) is as follows. Starting with λ add 1to the first entry λ times where λ is the unique element of the set { , . . . , p − } such that λ + ℓ µ X i =1 λ i ≡ p − . Proceed as in the case above clearing 1’s until all entries before position ℓ µ = n − n − . We then clear 1 from the entry in position n − λ ≤ p − p − n −
1) steps. Finally we add 1 and move it along into position s µ resulting in the weight M ( µ ). Combining this with the path from M ( µ ) to µ described above wehave a path from λ to µ that takes no more than ( p − n −
1) + P n − i =1 ( p − i = ( p − n − n )steps.Finally if µ ℓ µ = 0 and ℓ µ < n − λ to M ( µ ) is as follows. Starting with λ add 1 to the first entry λ times where λ is the unique element of the set { , . . . , p − } suchthat λ + ℓ µ X i =1 λ i ≡ p − . Proceed as in the case above clearing 1’s until all entries before position ℓ µ are 0 resulting in aweight whose entry in position ℓ µ is p −
1. Note that since ℓ λ ≤ ℓ µ the entry in position ℓ µ +1of the resulting weight is p − . We then subtract 1 from the entry in position ℓ µ and add it tothe entry in position ℓ µ +1 and repeat p − s ℓ µ resulting in the weight M ( µ ) . This path required at most ( ℓ µ +1)( p −
1) steps. Since ℓ µ < n − M ( µ ) to µ we have described a path from λ to µ thatrequires no more than ( p − n − n ) steps. This completes the proof. REFERENCES [1] G. Benkart, P. Diaconis, M.W. Liebeck, and P.H. Tiep,
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