NOTES ON MURPHY OPERATORS AND NAKAYAMA’S CONJECTURE MARK WILDON What follows are notes on G. E. Murphy’s paper The idempotents of the symmetric group and Nakayama’s Conjecture [3],readintheRepresentation Theory Advanced Class, Trinity 2008, Oxford. If you use these notes and have any corrections or suggestions for improvement, please let me know. 1. Residues and cores Givenapartitionλ, wedefinetheresidue ofthenodeinrowiandcolumn j to be j −i. We define its p-residue (also known as its p-class)to be its residue taken mod p. Early in §1 of [3], Murphy states the following non- obvious relationship between residues and p-cores, which later turns out to be critical to the success of his proof of the Nakayama conjecture. Proposition 1. Two partitions have the same multiset of p-residues if and only if they have the same p-core. Recall that the p-core of a partition λ is the partition obtained by re- peatedly removing rim-p-hooks from λ, until no more can be removed. In the diagram below showing (6,3,3,1), the 2-core is hatched, and nodes of 2- residue 0 are shaded. The thick lines show two of the three 2-hooks that can be immediately removed; the thin lines show the remaining three 2-hooks we remove en route to the core. Figure 1. The 2-core of (6,3,3,1). Assuming for the moment that the p-core is well-defined, it is quite easy to prove the ‘if’ direction of the proposition. For, if λ and µ are partitions of n with the same p-core, say γ, then we can obtain λ and µ from γ by repeatedly adding p-hooks. The result now follows from the observation that the set of residues of the nodes in any p-hook is always {0,1,...,p−1}. The ‘only if’ direction is harder. To prove it, we shall need G. D. James’ abacus notation for partition. Date: April 8, 2011. 1 2 MARKWILDON Review of abacus notation. Let λ be a partition of n. Starting in the southwest corner of the Young diagram of λ walk along its rim, heading towards the northeast corner. For each step right, put a space, indicated ◦, and for each step up, put a bead, indicated •. For example, the partition (6,3,3,1) has sequence ◦•◦◦••◦◦◦•. It is useful to allow such a sequence to begin with any number of beads, and to finish with any number of spaces; thesemustbestrippedoffbeforethepartitionisreconstructedfromitsbead sequence. One then arranges the bead sequence in p columns, known as the runners oftheabacus. Forinstance(6,3,3,1)isrepresentedona2-abacusasfollows. ◦ • ◦ ◦ • • ◦ ◦ ◦ • The reader should have little difficulty in seeing that the 2-hooks in (6,3,3,1) correspond to the beads on the above abacus having a space im- mediately above them. More generally, the north-eastern-most node in a rim-p-hook of λ corresponds to a bead on a p-abacus display for λ with a space immediately above it. An abacus display for the partition obtained by removing this hook may be obtained by sliding the bead one step up. It follows that the p-core of λ is obtained by pushing all the beads in an associated abacus up as far as they will go. The abacus makes it obvious that the p-core we reach is independent of the manner in which we remove hooks, and hence that the the p-core of a partition is well-defined. For example, (6,3,3,1) has the 2-core (2,1), shown in the abacus below. • • ◦ • ◦ • ◦ ◦ ◦ ◦ Remainder of proof. To prove the ‘only if’ part of Proposition 1 we need to show that a p-core is determined by its multiset of p-residues. (Thanks to the triangular shape of 2-cores, this is obvious when p = 2, but it is already non-obvious when p = 3.) The following proof is adapted from that of Theorem 2.7.41 in [1]. Let γ be a p-core. We may represent γ on a p-abacus using a multiple of p beads, say rp in all. Thanks to this convention, if we label the runners 0, 1, ..., p−1, then a bead on runner i corresponds to a step up past a removable node of residue ≡ i (mod p). For example, the 3-core (6,4,2,1,1) may be represented on the 3-abacus shown below. • ◦ • • ◦ •* ◦ ◦ • ◦ ◦ • 3 Thestarredbeadonrunner2correspondstotheremovablenodeofresidue2 in row 3 and column 2. By deleting rows of beads at the top of the abacus, we may assume, as is the case above, that at least one runner is empty. Now we repeatedly remove removable nodes from γ, until we reach the empty partition. Removing a node of residue ≡ i (mod p) corresponds to moving a bead one space left, from runner i to runner i − 1. (With the obvious modification that if i = 0 then i−1 is taken to be p−1.) If there are x nodes of residue i then this manoeuver must occur exactly x times. i i Bythetimewereachtheemptypartition, whichisrepresentedbyanabacus with exactly r beads on each runner, we have moved x beads from runner i i, and x beads to runner i. Hence, if we started with c beads on runner i+1 i i then x −x = c −r (indices taken mod p). i i+1 i Obviously, all the c are non-negative, and, by our choice of abacus display, i at least one is zero. Thus, given the x , we may use the last equation to i uniquely determine the c , and hence recover the p-core γ. This completes i the proof. 2. Murphy operators The Murphy operators L , for u ∈ {1,...,n}, are defined by u L = (1,u)+(2,u)+···+(u−1,u) ∈ ZS . u n It is easy to see that the Murphy operators commute; in fact L commutes u with any element of S . (We shall see very shortly, that the vanishing u−1 of L is due to the fact that 1 can only appear in a standard tableau in a 1 node of residue 0.) It is increasingly clear that the subalgebra of ZS generated by the Mur- n phy operators plays a critical role in the representation theory of the sym- metric groups—this subalgebra appears to be analogous in many ways to a Cartan subalgebra of a complex semisimple Lie algebra. For example, by Lemma 4 below, any QS -module decomposes as a direct sum of weight- n spaces for the L . u Givenatableaut,andanumberubetween1andn,letres(t,u)denotethe residue of the node of t containing u. The first unmistakable sign that the i Murphy operators are of interest is the following proposition, which shows that they can pick out these residues. In it we use the total order < defined on the set of row-standard tableaux of a fixed shape by setting s < t if the greatest number that appears in a different place in s to t appears higher up in s than in t. Proposition 2. Let t be a standard λ-tableau. Then e L = res(t,u)e +e t u t <t where e denotes a Z-linear combination of polytabloids e for tableaux s <t s such that s < t. This proposition is proved in Murphy’s earlier paper [2]. (In fact Murphy proves the stronger result that has e(cid:1)t with e<t in the above.) Here is an 4 MARKWILDON example, intended to give to illustrate the way in which Garnir relations are used in Murphy’s proof. Take 1 2 3 7 t = 4 6 9 . 5 8 We consider the effect of L on e . Easiest are the actions of (28) and (68), 8 t for it is immediate from the definition of the polytabloid e that e (28) = t t e (68) = −e . t t Nextweconsidertheactionof(38). Tohelpwiththiscalculation,weshall usethe(potentiallyhighlymisleading)shorthandthatatableautstandsfor its associated polytabloid e ; accordingly, we must write t 1 2 8 7 1 2 8 7 4 6 9 = − 4 3 9 5 3 5 6 as a sum of standard polytabloids. An application of the Garnir relation permuting the entries {2,3}∪{4,5} gives 1 2 8 7 1 3 8 7 1 3 7 8 1 2 8 7 4 3 9 = − 2 4 9 + 2 5 9 + 3 4 9 5 6 5 6 4 6 5 6 1 2 8 7 1 4 8 7 − 3 5 9 + 2 5 9 . 4 6 3 6 A further application of the relation permuting the entries {8,9}∪{7} com- pletes the rewriting. Note that if t(cid:48) is a standard tableau in the resulting expression then t(cid:48) < t. (For the general result behind this, see Lemma 2.1 in [2].) To find the action of the remaining summands of L , we consider the 8 Garnir relation permuting {1,4,5}∪{8} in the original tableau t. It gives e = e (18)+e (48)+e (58). t t t t Hence, adding up are results obtained so far, we find that e L = e −2e +e = res(t,8)e +e , t 8 t t <t t <t exactly as Proposition 2 predicts. 3. Idempotents Key to Murphy’s proof is the following construction of a complete set of primitive idempotents in QS . This first appeared in his earlier paper [2]. n Theorem 3. Let λ be a partition of n. Let d = dimSλ and let t ,...,t be 1 d the standard λ tableaux as ordered by the < order on tableaux. Let n−1 (cid:89) (cid:89) Lu−c E = ∈ QS . i n res(t ,u)−c c=−n+1 u i res(ti,u)(cid:54)=c TheE lieintheblockofQS correspondingtotherepresentationSλ. More- i n over, they form a complete set of primitive orthogonal idempotents lying in this block. 5 This result, despite its apparent complexity, follows quite easily from Proposition 2. We start its proof by showing that Sλ decomposes as a direct sum of weight-spaces for the algebra generated by the Murphy operators. Lemma 4. There is a basis {f ,...,f } of Sλ on which the Murphy opera- 1 d tors act by f L = res(t ,u)f . Up to a scalar, f is uniquely determined by j u j j i the equation f = e +e . i i <i Proof. Since two standard tableaux with the same residues are equal, it is possible to find x ,...,x ∈ Q such that 1 n x res(t ,1)+...+x res(t ,i) 1 i n n takes d different values as i varies from 1 to d. Now consider the linear map T = x L +...+x L ∈ End(Sλ). 1 1 n n With respect to the basis e ,...,e , we have 1 d   y 1 (cid:63) y2    T = (cid:63) (cid:63) y3   ... ... ...  (cid:63) (cid:63) ··· (cid:63) y d where y = x res(t ,1)+...+x res(t ,d). Now, by basic linear algebra, we i 1 i d i may find a unique y -eigenvector f for T of the form e +e . As T has i i i <i distinct eigenvalues, and the L preserve the T-eigenspaces, the vector f is u i a common eigenvector for the L . (cid:3) u We can now prove Theorem 3 by calculating the action of the E on the i basis f . Note first of all that, since the E are polynomials in the L , i i u they preserve the weight-space decomposition given by Lemma 4. However, if i (cid:54)= j, then we may find v ∈ {1,...,n} such that res(t ,v) (cid:54)= res(t ,v). j i HenceL −res(t ,v)isatermintheproductdefiningE , anditfollowsfrom v j i Proposition 2 that e E = e . Therefore f E = 0 if i (cid:54)= j. Similarly, one j i <j j i canshowbydirectcalculationthate E = e +e ,andthereforef E = f . j j j <j j j j Thus, assuming that the E do, as claimed, lie in the block of QS corre- i n spondingtoSλ, wehavefounddimSλ orthogonalidempotentsinthisblock. By dimension counting, they form a complete set of primitive idempotents. There is no particular difficulty in filling this gap in the proof, but as we havetoconsidere ,f andE definedwithrespecttodifferentpartitions,the j j i proof is inevitably slightly more involved; we refer the reader to Murphy’s paper [2] for the details. Notation change: From now on we will have to consider several different partitions at once, so we write Eλ rather than E for the idempotent just i i defined, and similarly we decorate the basis vectors eλ and fλ, and the i i tableaux tλ. i Remark. Let R be the multiset of residues of the partition λ. The denom- λ inator of Eλ is then i (cid:89) (r−c) r,c 6 MARKWILDON where the product is over all r ∈ R and c ∈ {−n+1,...,n−1} such that λ r (cid:54)= c. Note that this depends only on the partition λ, and not on which tableaux tλ we choose. A generalisation of this seems to be important to i the modular case. 4. Some calculations for (n−1,1) It is interesting to see the form of these idempotents in the case where λ = (n−1,1). In this case there are n−1 standard tableaux; when n = 4 they are 1 3 4 1 2 4 1 2 3 t = , t = , t = . 1 2 3 2 3 4 Note that t has i + 1 in its bottom row, not i. (This is annoying, but i essential if we are to be consistent with the notation used so far.) Usingmagmatohelpwiththecalculations(codeavailableatthewebpage http://www-maths.swan.ac.uk/staff/mjw/other.html), one finds that: (1,1) E = 1 − 1(12) 1 2 2 (2,1) E = 1 − 1(12)+ 1(13)+ 1(23)− 1(123)− 1(132) 1 3 3 6 6 6 6 (2,1) E = 1 + 1(12)− 1(13)− 1(23)− 1(123)− 1(132) 2 3 3 6 6 6 6 (3,1) E = 1 − 1(12)+ 1 (13)+ 1 (14)+ 1 (23)+ 1 (24)+ 1(34) 1 8 8 16 16 16 16 8 − 1 (1234)− 1 (1243)− 1 (1342)− 1 (1432)− 1(12)(34) 16 16 16 16 8 − 1 (123)− 1 (132)− 1 (124)− 1 (142) 16 16 16 16 + 1 (134)+ 1 (143)+ 1 (234)+ 1 (243) 16 16 16 16 (3,1) E = 1 + 1(12)− 1 (13)+ 5 (14)− 1 (23)+ 5 (24)+ 1 (34) 2 8 8 16 48 16 48 24 − 1 (1234)− 1 (1243)− 1 (1324)− 1 (1342)− 1 (1423)− 1 (1432) 48 48 12 48 12 48 + 1 (12)(34)− 1 (13)(24)− 1 (14)(23) 24 12 12 − 1 (123)− 1 (132)+ 5 (124)+ 5 (142) 16 16 48 48 − 1 (134)− 1 (143)− 1 (243)− 1 (234) 48 48 48 48 (3,1) E = 1 + 1(12)+ 1(13)− 1 (14)+ 1(23)− 1 (24)− 1 (34) 3 8 8 8 24 8 24 24 − 1 (1234)− 1 (1243)− 1 (1324)− 1 (1342)− 1 (1423)− 1 (1432) 24 24 24 24 24 24 − 1 (12)(34)− 1 (13)(24)− 1 (14)(23) 24 24 24 + 1(123)+ 1(132)− 1 (124)− 1 (142) 8 8 24 24 − 1 (134)− 1 (143)− 1 (234)− 1 (243). 24 24 24 24 (3,1) It is interesting that the numbers in the denominator of E are all powers 1 (3,1) (3,1) of 2, while factors of 3 appear in E and E . (This does not contradict 2 3 the remark at the end of §3, because the numerators of the Eλ certainly i (3,1) do depend on i.) The curious fraction 5/48 in E is also worth noting. 2 (n−1,1) (n−1,1) Taking the sum E + ... + E we obtain the primitive central 1 n−1 7 idempotent in QS for S(n−1,1), n n−1 (cid:88) z(n−1,1) = (|Fixσ|−1)σˆ. n! σ Here, σ runs over a set of representatives for the conjugacy classes of S , n and σˆ denotes the sum of all elements conjugate to σ. For example, z(1,1) = 1 − 1(12) 2 2 z(2,1) = 2 − 1((cid:100)12) 3 3 (cid:92) (cid:92) z(3,1) = 3 + 1((cid:100)12)− 1(12)(34)− 1(1234) 8 8 8 8 Using these calculations, we can give examples of some of Murphy’s other results. For example, his Lemma 1.8 in [3] predicts that if u − 1 and u are in the same row or column of t , then (u − 1,u) commutes with E . i i In particular, every E commutes with (12), as can easily be verified for i the idempotents above. (This can also be seen in another way: every L u commutes with (12), and the E are polynomials in the L .) i u References [1] James,G.,andKerber,A.Therepresentationtheoryofthesymmetricgroup,vol.16 ofEncyclopedia of Mathematics and its Applications.Addison-WesleyPublishingCo., Reading, Mass., 1981. [2] Murphy, G. E. A new construction of Young’s seminormal representation of sym- metric groups. J. Alg. 69 (1981), 287–297. [3] Murphy,G.E.TheidempotentsofthesymmetricgroupandNakayama’sconjecture. J. Alg. 81 (1983), 258–265. E-mail address: [email protected].