259

The Potts model and the symmetric group

V. F. R. JONES Dept. of Math., University of California Berkeley Berkeley, CA 94720, USA

ABSTRACT The symmetric group S k acts on a vector space V of dimension k by permuting the basis elements VJ,v2,· .. vk. The gmups 5 11 acts on ®n V by the tensor product factors. We show that the algebra of all matrices on 0nV commuting with Sk is generated by Sn and the operators e 1 and e2 where 1 k el ( VPl 0 Vp2 0 ... 0 Vp.,) = k L Vi 0 Vp2 0 ... 0 Vpn i=l

The matrices e 1 and e2 give the vertical and horizontal transfer matrices adding one site in the square lattice Potts model.

§0. Introduction In the Potts model ([Ba]) of classical equilibrium statistical mechanics, indi- 11 vidual "atoms" can be in one of k possible spin" states o-1 , ..• , an. Two particles interact if they are neighbours and if their spins are the same. Thus the matrix w(u, a') of Boltzmann weights is invariant under the (diagonal) action of the metric group Sk. The usual presentation of the Potts model transfer matrices in the physics literature ( eg. fBaJ) writes them as elements of an algebra gener- ated by cyclic groups so only the cyclic symmetry is apparent. This is a little misleading and one should attempt to understand the full symmetry group. This seems especially important now that there are known to be solvable models whose Boltzmann weights have rather interesting symmetry groups (see [JaJ). The first question one may ask is to what extent the transfer matrices generate the commu- tant of the group of symmetries. The actual details of this question will depend on 260 the kind of model (Spin, vertex, IRF, ... ). Although we are only really concerned with the Potts model here, let us describe the situation for a certain vertex model (solvable by Temperley Lieb equivalence- see [Ba] chap 12) to illustrate clearly the connection between this problem and classical invariant theory. A vertex model is specified by a k2 x 12 matrix b = 1, ... k, x, y = 1, ... I. In our special case we define = -/;6,z6b,y (k = 1), and R(>.) = f(>.)e + id for some conveniently chosen J(>.). The site-to-site transfer matrices are then obtained from the matrices e1, e2, .•• , en-I on ®nV, V being a vector space over a field of characteristic zero, where ei{w1 ® · · · 0 Wi ®wi+I 0 · · · ® wn) = w 1 ® · · · ® e( Wi ® Wi+l) 0 · · · ® Wn, where Wi E V for each i and e is the element of End(V 0 V) having matrix with respect to the basis {v, 0 Vz} of V 0 V, {V,} being a basis of V. (Thus e(v, 0 Vz) = 2::: e: 0 Vy.) b,y The ei 's are orthogonal projections satisfying

1 eiei±1 ei = k 2 ei eiej = ejei if li Jl 2.

If V is considered to be a real Euclidean space for which {va} is an orthonormal basis, one recognizes e as being orthogonal projection onto the canonical subspace of V 0 V, scalar multiples of L Va 0 Va, which is independent of the orthonormal basis. Thus this vertex model" has as a natural symmetry group the orthogonal group O(k). The question of to what extent the site-to-site transfer matrices gen- erate the commutant of the symmetry group becomes the question of an explicit description of the commutant of the tensor powers of the vector representation of the orthogonal group. Thus question was answered in the 1930's by Brauer [Br] who showed that this commutru1t is generated as an algebra by our ei's and the natural representation of the permutation group Sn on ®n V. He gave moreover an explicit abstract algebra with basis all possible partitions of a set of 2n elements into subsets of size 2, represented by diagrams as below,

X X X X

X 261 and an obvious way of considering these diagrams as operators on ®nV. The ele- ment ei described above corresponds (up to a factor k) to the diagram

X X X X X (\ X X X X X X X 2 i +I n

For k sufficiently large (and n fixed) the matrices corresponding to diagrams are a basis of the commutant of O(k), and they always span it linearly. Thus the commutant is always a quotient of the abstract Brauer algebra. Wenzl obtained the structure of the abstract Brauer algebra for generic values of the parameter k, using ideas from subfactors, knot theory and quantum groups, in (VV]. (The interested reader might also want to consult [Jol].) Thus we see that, for this vertex model, the commutant of the symmetry group is generated by the site-to-site transfer matrices and the obvious action of the permutation group on @ 11 V. One may ask to what extent this is true in general. (For vertex models coming from quantum groups, the group symmetries would become quantum group symmetries and the group Sn would be replaced by the braid group - the result may well be true in general, through one would have to consider R(A) for all values of>..) In this paper we will show that the version of the result appropriate for spin models (i.e models of the same kind as the Potts model but with arbitrary Boltzmann weights w(a,a')) is true for the Potts model. The main differences are that @ 11 V will carry 2n matrices ei, that the parameter k2 in the ei relations becomes k, that the permutation group Sn has two orbits on the set of all ei's, and that the continuous group 0( k) of the above vertex model is replaced by the finite group sk. Before getting on with the proof of the result let us point out that the algebra structure {and thus, if desired, the dimension) of the cmpmutant of Sk is easily determined, at least inductively, from the representation: theory of Sk· For the representation on V of S1.: is a permutation representation, so tensoring by V cor- responds to restriction to the stabilizer of a point, i.e. Sk-I and then inducing back up to Sk. Since induction and restriction are particularly simply visible from the Young diagrams, one immediately obtains the Bratteli diagram for Endsk(0 71 V): (illustrated for k = 6): 262

- dimension 2

- dimension 15

- dimension 203

etc.

In general for a given k the Bratteli diagram will stop growing wider after the point at which all irreducible representations of Sk appear. In this respect the Brauer algebra is different the diagram will grow ever wider since the orthogonal group has infinitely many different irreducible representations. It is still true however that the number of simple components of the Brauer algebra is larger if k is larger (for large n). I would like to thank Roger Howe for first pointing out to me the ,en matrix in the vertex model described above.

§1. Two bases for Ends,(®"V)

Let V be a vector space of dimension k over a field ]{ of characteristic zero, with basis {va} indexed by a set A of size k. The symmetric group Skis linearly 11 represented on V by a(va) = V 11 (a). Thus for each n, ® V is an Sk-module. As a vector space, Endsk(011 V) is just the fixed points for the representation of Sk on End( ®n V) by conjugation. But this is a permutation representation for if

X E End( 0n V) has matrix with respect to the basis {Va 1 0va 2 0· · ·0Van} then a(X) has matrix Hence a basis for Endsk ( ®nV) is given by 263

where each sum is taken over all {a1, a2, ···an, b1, b2, · · · bn} in an orbit 2 orbit for the action of Sk on the 2n-fold Cartesian product A n. Each orbit for this Sk action is given by a partition of {1, 2, ... , 2n} into subsets (obviously at most k of them). If...... , denotes the equivalence relation defined by such a partition then the 211 corresponding orbit on A consists of these 2n-tuples ( ah ···,an,···, a2n) (setting b; = Un+i) for which ai = Uj iff i rv j. The number of equivalence relations with r classes is the Stirling number S(2n, r) so that the dimension of Ends, ( 0" V) is k 2n LS(2n, i). For 2n S: k this is the Bell number B(2n) = L S(2n, i) (see [S) p.33). i=l i=l Compare the first few Bell numbers with the dimensions of the Bratteli diagram. If .-...- is an equivalence relation on {1, ... , 2n} we will call T ..... the element of End(V®") defined by the corresponding Sk-orbit. It clearly has the following matrix: = { 1 if (ai aj ¢> i "'j) 0 otherwise. Note that T,., is zero if the number of classes for "' is more than k. i.,From our discussion {T...... I "' an equivalence relations } is a basis for Endsk(®nV) with k L S(2n, i) elements (forgetting any T_ that may be zero). i=l In fact we will use another set which linearly spans Endsk(®nV) and will also be a basis if k 2 2n. For each as above define L_ E Ends, ( ®" V) by

Where the sum is over all partitions "-' 1 coarser than ,....., . By MObius inversion ([S) p.ll6) the T_'s can be expressed in terms of the L_'s so they also span Endsk(®nV). Note though that the L .. ,/s are always nonzero so they will not form a basis for k < 2n. We leave the verification of the following lemma to the reader as it should make clear ·what is going on.

Lemtna 1. 1) is defined by i i + n then L_ = id. 2) is defined by i i + n fori > 1 tl>en

L ...... = ke 1 with e1 as in tile introduction. 3} is defined by 1 2 (n + 1) (n + 2) and i i + n fori> 2 then

L ...... = e2 with e2 as in the introduction. 264

§2. Turning L"' into a "planar" fortn Consider a rectangle with n marked points on the bottom and the same n on the top as in the figure (where n = 5)

Surround each of the marked points on the top and bottom with two close hours (marked with a X in the figure). Now join the points marked with "x'' to each other, within the rectangle, by any system of non-intersecting curves. The regions inside the rectangle can then ne shaded black and white (with the regions touching the left and right sides of the rectangle shaded white. Any such diagram defines a partition of the original 2k marked points. VVe will call such a partition (or it's equivalence relation) '1planar". Note also that a planar partition completely determines up to isotopy ( rel the boundary) the system of curves joining the points marked "x''. It is well known and easy to prove by induction that the number 1 4 of planar pa.rtitions is the Catalan number ---( The connection schemes 2n + 1 2nn). of the points marked "x, were exploited by Kauffman in his ''states modeP' for a knot polynomial (see [K]). The result of this paper will follow from the following lemma which is almost obvious.

Letnma 2. If,....., is any equivalence relation on the 2k marked points in a rectangle

as above, one may choose 2 permutations o- 1 and o- 2 of tl1e top k and bottom k points respectively, so that ,....., becomes planar upon reordering the top and bottom points according to a: 1 and a:2.

Proof Begin with the bottom points and look at those equivalence classes for ,...., which contain only bottom points. Choose o- 2 so as to set all these points to the left. Do the same with the top and n 1 • Now choose an equivalence class for "' involving both top and bottom points and extend a:1 and n 2 so that these points all occur, with no gaps, to the right of the points already decided upon. Continue 265

until exhaustion. The resulting partition will have the form shown below and is

obviously planar:

Q.E.D.

§3. Proof of the theorem.

The diagrams joining the 2n bottom x's to the 2n top x's form the basis of

an algebra sometimes called the Temperley Lieb algebra. It can be thought of as

a subalgebra of the Brauer algebra defined in the introduction as each diagram

gives a partition of the 4n x's into subsets of size 2. The multiplication is defined

by a 2-step procedure.

Step 1. Stack the two rectangles on top of each other, lining Up the x 's.

Step 2. Remove the middle edges and middle x 's. One then has a new diagram

possibly containing some closed loops. If there are p closed loops, the product is then the resulting diagram, with the closed loops removed, times the scalar OP where 0 is some fixed element of the field. One thus obtains a family of algebras 1 4 K(2n, 8) of dimension ---(n). We illustrate multiplication inK( 4, .5) below. I 2n + 1 2n 266

-0--- - so cx(J = 8 =

(3=

We now define special elements Et, E2, ... E2n-I 'of K(2n, 6): v

E,

Ef = 8E; One easily has E;E = EjE;, fli- jl 2 2 { 1 EiE1±1Ei = Ei and it is easy to see that the Ei's, together with 1, generate K(2n, 6). Counting reduced words on the Ei's one obtains the Catalan numbers {see [Jo2]) so the above relations present K(2n, 6).

Lenuna 3. If d is a diagram giving a basis element of I<(2n, 0) as above, let I"Jd be the equivalence relation it generates. Then the map

l odd 1. even extends to an algebra lwmommphism from I<(2n, f) to End(0"V) so that for a diagram d,

Proof Since the above relations present K(2n 1 jJ, the existence of r.p follows from the fact that the r.p(Ei) satisfy the same relations. We leave it to the reader to check that if cl and d1 are diagrams then L ..... dL-d' is a multiple of L-dd'· Q.E.D. Remark. It seems suggestive that the mapping r.p defined above is injective as sooon as k 4 whereas we have seen that for partitions there are linear relations between the L..., as soon as n > k. The same thing happens with the I<(2n, 8) as a subalgebra of the Brauer al- gebra. This linear independence is easily proved using the Markov trace technique of [Jo2].

Theorem. The commutant of Sk on ®nV is generated by Sn, e1 and e2 where V, e1 and e2 are as in Lemma 1 and Sn acts by pennuting the tensor factors.

Proof. It suffices to show that any L""' is in the algebra generated by Sn and e1, e2. We have seen that by multiplying L ..... on the left and right by elements of Sn, we can assume that "' is planar. The elements tp(E3 ), tp(E.1), ••. tp(En) are just conjugates of tp( E1) and tp( E2 ) under the action of 5 11 so the theorem follows from Lemma 3. Q.E.D.

References

[Ba] R. Baxter, "Exactly solved models in statistical mechanics'', Academic press, London, 1982. [Br] R. Brauer, "On algebras which are connected with the semisimple continuous groups", Ann. Math. 38 (1937) 854- 887. [JaJ F. Jaeger, "Strongly regular graphs and spin models for the Kauffman poly- nomial". Geom. Dedicate 44 (1992) 23- 52. [Jol] V. Jones, "Subfactors and knots", CBMS series number 80 (1991). [Jo2] V. Jones, "Index for subfactors", Invent. Math. 72 (1983) 1-25. [K] L. l(auffman, "State models and the Jones polynomial", Topology 26 (1987) 395 - 401. (S] R. Stanley, "Enumerative combinatorics", voll. VVadsworth and Brooks/Cole (1986). [W] H. Wenzl, "On the structure of Brauer's centralizer algebras", Ann. Math. 128 (1988) 173- 193.