The Bruhat order and iterated exponentials

JOURNAL OF COMBINATORIAL THEORY, Series A 9, 87-99 (1989)

The Bruhat Order and Iterated Exponentials

JOHN R. STEMBRIDGE*

Department of Mathematics, University of California,

Los Angeles, California 90024

Communicated by the Managing Editors

Received June 22, 1987

A partial order on the symmetric group was recently defmed by B. W. Brunson, based on a hierarchy of iterated exponentials. Several properties either proved or

conjectured by Brunson strongly suggest that this partial order is equivalent to the Bruhat order. In this paper, we give a simple, explicit description of the order which disproves the conjectures and shows that, although it is not the Bruhat order, it has a strikingly similar characterization. 6 1989 Academic Press, Inc.

1. INTR~OUCTION

A partial order on the symmetric group S, was recently defined by Brunson [Br], based on a hierarchy of iterated exponentials of real variables. A number of interesting properties of the order were proved by Brunson, although an explicit description remained elusive. Indeed, to prove that two permutations are related in this order (hereafter referred to as the “exponential order”) would amount to proving formidable inequalities involving iterated exponentials.

A direct (but not explicitly stated) consequence of Brunson’s work is the fact that the exponential order of S, contains the Bruhat order of S, (defined below). In fact, the diagrams supplied by Brunson show that for n d 4, the Bruhat and exponential orders on S, coincide. Moreover, Brunson formulated two conjectures concerning the structure of the exponential order, both of which are satisfied by the Bruhat order. This immediately suggests a third conjecture; namely, Bruhat = exponential.

The Bruhat orders arise naturally in a number of areas, such as the geometry of Schubert cells, the combinatorics of Coxeter groups, and statistics (see, e.g., the excellent bibliography of [Bj]). In spite of this diversity, the validity of the exponential/Bruhat conjecture would yield a remarkable characterization of the Bruhat order unlike any previously known.

* Partially supported by NSF Grant DMS-8603228. Current address: Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1003.

87 0097-3165/89 $3.00

Copyright 0 1989 by Academic Press, Inc. All rights of reproduction in any form reserved.

88 JOHN R. STEMBRIDGE

In this paper, we will derive a simple description of the exponential order. This description will show that, unfortunately, all of the above conjectures are false for n 2 5. However, despite these negative results, we will find that the exponential and Bruhat orders do share a remarkable property. Both orders may be described as the intersection of certain naturally associated “maximal quotients,” in a sense to be made precise in Section 4.

2. THE EXPONENTIAL ORDER

For any positive real variables x,, x2, . . . . x,, let x1 r x2 T . . . T x, denote the corresponding tower of iterated exponentials evaluated from the left; i.e.,

where x T y = xy. The exponential order arises from the problem of deciding when an exponential tower, constructed from some permutation of the variables xi, . . . . x,, is larger than another. Thus, a typical problem might be to decide whether

given that x > y > z. Of course, if there is no further information regarding the variables x, y, z, it is unlikely that any nontrivial prediction could be made. Indeed, one cannot decide whether xy > y* given only that y > x > 0. However, since x/log x is increasing for x > e, we have

XTY>YTX for y>x>e.

Therefore, in order to enhance the likelihood that nontrivial predictions can be made in problems of the form in (1 ), it is natural to impose the restriction x > y > z 2 e.

A crucial idea that seems contrary to initial expectations is the realization that this hierarchy of iterated exponentials becomes easier to analyze when the variables are allowed to repeat within the towers. For example, anticipating the description of the exponential order in Theorem 3.1, one may prove that

for y ax 2 e. Thus, a more natural domain for the exponential order consists of n-tuples of variables with repetitions allowed, rather than the permutations originally proposed in [Br].

ITERATED EXPONENTIALS 89

The preceding ideas motivate the following definition of the exponential order. For any positive integer k, let [k] = { 1,2, . . . . k}. A word u of length n over the alphabet [k] is simply an n-tuple u = u1 u2 ... u, E [k]“. There is a natural way to associate with u a tower (u) of n iterated exponentials in the variables x,, x2, . . . . xk. Specifically, we define

(UXX,, ..., &I = x,, t xtq t . . . t X,“.

The exponential order is the partial order < E on [k 1” defined by

U<,V iff <u>(x) G (v>(x) for all xET,,

where r,={x~R~:x~~x~~...~x~~e}. The fact that this relation is antisymmetric may be seen by considering choices XE r, for which x1, x2, . . . . xk are sufficiently far apart. Alternatively, one may consider the remark following Theorem 2.2 below.

Our first step in determining an explicit description of the partial order cE is to examine the local behavior of exponential towers near points of the form (a, a, . . . . a). Although it is far from obvious, these neighborhoods contain sufficient information to determine the exponential order.

LEMMA 2.1. Assume xi > 1 for i > 1:

(a) Letf,=x,f-..tx,fx,. We have

(b) Let u~[k]“. Let (u)~=u,-~+~...u, denote the suffix of u of length j (1 <j < n). We have

aw -=& C,iOg((u),_i+,)...log((u),) (1GW. axi , 1 “,-I

Proof: For part (a), notice that f, = x, tf, _ r. Therefore

if i=n.

if i< n,

and the result follows by induction on n. Part (b) follows from (a) by the chain rule. a

Let < L denote the lexicographic order on (0, 1 }” defined by insisting


FIGURE 1

that u cr. v whenever there is an index i (1 < i< n) such that ui < vi and uj=vj forj>i. For example, OOO<. 100<,OIO<L llO<,OOl cL 101 cL 011 <L 111.

Define the quotient order -cB to be the following partial order on (0, 1)“:

Ul <Q vl, UO <Q vo if ucLv @a)

UOO<eVO1,UOO<,vll,ulO<,vll for all U, DE (0, l}“-’ (2b)

UlO <Q 001 if u<.v. (2c)

For n < 2, the lexicographic and quotient orders coincide. The Hasse diagrams of <o for n = 3 and n = 4 are given in Fig. 1 (rotated 90” from the usual convention). Notice that the change of alphabet 0 + 1, 1 -+ 0 induces an anti-automorphism of both < L and < o.

For any i (1 <i<k) there is a natural map cpi: [k-J”+ (0, l}” induced by the change of alphabet { 1, 2, . . . . i) + 1, {i + 1, . . . . k} + 0. For example, (~~(2143551) = 1101001. We claim that these maps are order-preserving; i.e.,

THEOREM 2.2. rf u, DE [k]“, then u<, u implies vi(u) <Q q,(u) (1 sick).

Proof Assume u GE u and let 1 Q i < k. Consider the behavior of (u)(x) and (v)(x) along trajectories in r, of the form

x(t) = (a, a, . ..) a) + t( 1, . . . . 1, 0, . ..) 0)

for any fixed a 2 e. We have (u)(x(O)) = (v)(x(O)) = a,, where

ITERATED EXPONENTIALS

Therefore, u d E u implies

91

Xu>(x(t)) at

~ a(o>(x(t)) r=0 at . I=0

By Lemma 2.1(b), we have

and similarly for u. Therefore,

1 l"gan-j+l ..*loga,< c loga,~j+l...log.n (3) “, c i L> C i

for every a > e. Consider the sequence (~l,)~ 5 o defined by 01,: ’ = log a 1 . . . log uj (a0 = 1).

Note that for any 1> 0, we have

jFIaj$a, 1 ~-‘=~<a,. j>O u-l

Since c~i = cl,/log a, the above estimates fail for I= 0. However, in case log a > 2, we have

c a,=ct,+ c orj<2c(,=2/loga<1 (=c10). jz0 .i> 1

It follows that if (a,) and (clq) are two (possibly finite) subsequences of (aj) with 0 < rl -c rz < . . . and 0 <s, < s2 < ‘. . , then

c %,-c c NJ, (4) j> 1 i> 1

if and only if (rj) follows (sj) lexicographically in the sense that the minimum j for which rj # sj satisfies rj > 3,.

The relevance of these calculations can be seen when (3) is renormalized by a factor of CI,. The result is a relation between subsequences of (aj); namely,

C a,jSolIian-j. (5) 4-s I I-

It follows that q,(u) <r. q,(o). We may immediately deduce cpi(u)GQ vi(u) unless q,(u)= z’il0 and

q,(u) = 601 for some 6, BE (0, l}n-2 such that li >L t?. In that case, consider the inequality (5) at a = e. The lexicographic characterization in (4) breaks


down, but may be salvaged for subsequences (a,) and (a,) of (M~)~, 1. Since a0 = a, = 1 for a = e, we see that ti >L 6 would contradict (5), thus proving cPiC”) GQ cPil”). I

We remark that the antisymmetry of cE can be deduced from Theorem 2.2. If u <E u and u <<E U, then we have cpi(u) = P,(U) for 1 <i< k; these conditions clearly force u = v.

3. THE MAIN THEOREM

The main result of this paper is the following theorem, which shows that the converse of Theorem 2.2 holds, and thus yields a description of the exponential order.

THEOREM 3.1. Let u, UE [k]“. We have u<,u iff q,(u) <a q,(u) (1 <i<k).

Before proving the result, we consider two examples. Applying the two projections ‘p, and cpz to the words 123312 and 321213, we find

123312~ 100010, 110011

321213t-+001010,011110.

Since 100010<,001010 and 110011 >,OllllO, we conclude that 123312 and 321213 are unrelated by < E. (This follows even from Theorem 2.2.)

Next, consider the permutations 43215 and 34512. Each permutation has four projections into {O, I)‘; namely,

34512~00010,00011, 10011, 11011

43215t-+OOOlO, OOllO,OlllO, 11110.

Since each of the corresponding projections are in Q-order, we conclude that 43215 cE 34512. The Hasse diagrams for the partial orders (S,, < J and (S,, -C E) appear in Fig. 2.

In the proof of Theorem 3.1, it will be convenient to use estimates of 8/8x,(log log(u)). If we rewrite Lemma 2.1(b) using the fact that

1 dy $loglogy=-----, ylwydx

we obtain


4312 4231 3421 321

312 231

HI 132 213

1423 1342 2143 3124 2314

FIGURE 2

LEMMA 3.2. Let UE [k]“. We have

&log log(u) = I

& C,log((u),~j+~)".log((u),-,). I 1 u,-’

We remark that although this would seem to be an innocuous refor- mulation of Lemma 2.1, the reader will find it challenging to prove the theorem using estimates only of 8/dxi(log( u)).

Proof of Theorem 3.1. We prove by double induction on n and k that for any pair of words U, u E [k]“, v,(u) <o q,(u) (1 <i< k) implies u GE u. When n = 1 or k = 1 the assertion is obvious, so we need only consider cases in which n, k 2 2. For the induction hypothesis we assume the result for all pairs of the form (n’, k) with n’<n, and for the pair (n, k- 1).

To prove the result for the pair (n, k), let U, u E [k]“, and suppose that q,(u) -<o q,(u) (1 d i< k). Regard log log(u) and log log(u) as functions of x, on the interval [x2, +a~). To prove that u<~u, it suffices to show that

<~)(XZ,X2,X3,...,Xk)~ (ux%x2,x3,...,-%) for (x,, . . . . xk) E r, - ,

(6)

&loglog(u% T&g10gw for xET,. (7) 1 1

For (6), let u’, u’ E [k - 13” be the words obtained from U, u by replacing all occurrences of j with j- 1 (2 <j < k). Since cp,(u’) = cpj+ 1(~) and qj(u’)=qj+r(u) for lgj<k-1, we have qj(u’)<o’p,(u’) (l<jck-1). By the induction hypothesis, it follows that U’ <E 0’; i.e., (u’)(x) ,( (u’)(x)


for all x E r, _, . In particular, we may deduce (6), since ( u)(xz, x2, x3, . . . . xk) = (u’)(x,, x3, . . . . x,), and similarly for u.

For (7), we first observe that the quotient order is preserved when suf- fixes are taken; i.e., if w, w’ E { 0, 11” satisfy w Go w’, then ( w)~ <Q ( w’)~ for 1 <j<n (using the notation defined in Lemma 2.1). Since qi(u)de q,(u) for 1 < i < k, we also have cp,( (u),) de cp,( (a),), so by the induction hypothesis, (u!,<~(u)~ for 1 <j<n.

For brevity, define L,(u)=log((u),-j+,) . ..log((~).~~) and L,(u)= 1. We have Lj(u)(x) < L,(u)(x) whenever x E r, and 1 <j < n. For any fixed r (1 <r<n), we claim that CIGjCr L,(u) < L,(u). To see this, observe that

+ w%((~)“-r+,, log((uLr+2))-’ + .‘. 1

<L,(u) $+ [

1 + . . . en-ren-r+l

<L,(u)(e’1iep2+~..)<L,(u). 1

We remark that the constraint r < n is essential, since the estimate fails when r = n.

Let A #B be subsets of [n], and define m = max(A n B), where A n B denotes the symmetric difference. Assume m E B. Using the above estimates, it follows that

1 L,(u) < C LJu) for all XE r, jeA jsB

whenever any of the following conditions are satisfied: (a) m < n, or (b)m=nandn-14A-BB,or(c)m=n,n-1EA-B,andBcontainsthe largest member of A n B smaller than n - 1, if such members of A n B exist. (For (b) and (c), we are implicitly using the fact that L, _ 1(u) < L,(u), which is a consequence of the lower bound log xi 2 1.)

By Lemma 3.2, we have

~loglog<u)= 1 x1 lctgx,JALj@)

and

~loglog(u)= I XI lolg X1 TB -Vv),

where A = (j: uj= l} and B= {j: vi= l}. Since cpl(u)<o q,(u), one of the relations (2a), (2b), or (2~) must apply to the words qpl(u) and cpl(u).


However, these conditions are equivalent to the above cases (a), (b), and (c), respectively. This proves the inequality (7) and completes the induction. 1

Recall that the change of alphabet 0 c1 1 induces an anti-automorphism of the quotient order on (0, 1 }“. Consequently, we have

COROLLARY 3.3. The map i --+ k + 1 - i induces an anti-automorphism of < E. In particular, < E is self-dual.

We remark that the exponential towers in [Br] are restricted to the domain e<x,<x,<...Qx,. Therefore, the partial order (S,, < ,J and Brunson’s exponential order have different labelings, although they are isomorphic. By the above corollary, Brunson’s labeling may be obtained by turning (S,, < E) upside down.

4. A COMPARISON OF THE BRUHAT AND EXPONENTIAL ORDERS

Henceforth, the domain of < E will be restricted to S,. For any u E S,, let inv(u) = I{ (i,j): 1 Q i <j< n, U, > ui} 1 denote the number of inversions in u. The Bruhat order of S, is the partial order <B obtained from the transitive closure of the following relation:

u<t.u whenever inv(u) < inv( t . u), #ES,,, tET,

where T = { (ij): 1~ i <j < n} denotes the set of transpositions in S, and t . u denotes the usual group multiplication. It is well known that cB is ranked and that inv serves as the rank function; i.e., any maximal chain from ZJ to u in < B has length inv(u) - inv(u). The diagrams of the Bruhat orders of S, and S, appear in Fig. 2, since < B = < E for n < 4, as the reader can easily verify.’

The following result is implicit in [Br].

PROPOSITION 4.1. Zf u, v E S,, then u Ge v implies u GE v.

Proof. By Theorem 3.1, it suffices to show that if t E T and inv(u) < inv(t .u), then (Pi <o qk(t. U) for 1 <k < n. Suppose that t = (ij), where i <j. The condition inv(u) < inv(t . U) implies that i occurs to the left ofj in U. If k < i or k 2 j, then (Pi = cpk(t. U) and there is nothing to prove. If i < k <j, then qk( t . U) is obtained from (Pi by exchanging a 1 and a 0 in (Pi with the 1 moving to the right. By the definition of the quotient order, it follows that (P,+(U) <o cpk(t. u). 1

L This tedious exercise can be avoided by using a technique to be discussed later.

58?a/50/1-7


If < is any partial order, use the notation G to denote the covering relation. Thus, u -G u iff u < u and there is no w such that u < w < u. The following result shows that, at least for some choices u, u E S,, it is possible to strengthen Proposition 4.1 and prove that u -Go u implies u -Go u (cf. [Br, Theorem 51).

PROPOSITION 4.2. If t = (ii + 1) is an adjacent transposition, then inv(u) < inv( t . u) implies u GE t .24.

Proof If inv(u) < inv(t . u), then i occurs to the left of i -)- 1 in u, and t. u is obtained from u by interchanging i and i + 1. Thus, (PJu) = cpk(t. u) for k # i. If u <E v <E t .u, then q,Ju) <o qk(v) <o qk(t. u) by Theorem 3.1, so (PJu) = q,Jv) for k # i. Therefore, u and u must agree except possibly at the positions occupied by i and i + 1, so that u = u or t u = u, i.e., t . u covers 24. 1

We remark that the transitive closure of the relation that appears in Proposition 4.2 (u < t. u if t = (ii + 1), inv(u) < inv(t . u)) is known as the weak order of S, [Bj].

These propositions, the coincidences for n < 4, and the two conjectures proposed by Brunson naturally lead one to suspect that the exponential and Bruhat orders are the same. However, using the characterization of < E provided by Theorem 3.1, it is a simple matter to show that the exponential/Bruhat conjecture and Brunson’s conjectures fail for n 2 5.

Specifically, from the example in Section 3, we know 43215 cE 34512. Since inv(43215) =inv(34512) =6, it follows that 43215 and 34512 are unrelated in < B, so < E is a proper extension of cB for n = 5. It is easy to fit this example inside larger symmetric groups and thereby conclude that < E# cB for n 2 5. Using the subword property of the Bruhat order [SW], it is not hard to see that Conjecture A of [Br] would imply, if true, that u Q, v implies u Q~V. However, we have 34215 ~~43215 cE 34512 so that 34512 does not cover 34215 in cE. Since inv(34215)= 5, inv(34512) = 6, and 34215 cB 34512, it follows that 34215 ~~34512, which disproves Conjecture A. The permutations 43215 and 34512 also disprove Conjecture B of [Br], which asserts that inv is a rank function for (S,, <El.

We mention that Brunson states without proof a result [Br, Theorem 61 similar to Proposition 4.2; namely, that if t = (ii + 1) and inv(u) c inv(u . t), then u <E u. t. However, the permutations 23145 and 23415 provide a counterexample (i.e., take u=23145, t= (34)). We have inv(23145) < inv(23415), but 23145 cE 32145 cE 23415, which shows that u-t does not cover u in this case.

On the positive side, there is a surprising connection between the Bruhat and exponential orders that arises from the fact that Theorem 3.1 is a direct


analogue of a characterization of the Bruhat order due to Deodhar [D]. That is, there is a partial order on (0, l>“, denoted by co, with the property that

U<,V iff cp,(u) 6 cp,(u) for ldi<n.

To describe this partial order, we first observe that for any u ES,, the projection q,(u) is a word with i l’s and n - io’s. Hence, it would be equivalent to describe a family of partial orders of the form ((0, 1 >rv < B)( 1 d i < n) such that

uQeu iff cpdu) <B cp,(o) for ldicn, (8)

where { 0, 1 }; denotes the set of words consisting of i l’s and n - i 0’s. The partial order <B may be obtained by restricting the Bruhat order to a collection of distinguished coset representatives for the quotient S,/Sr), where S:) z Si x S,- i is the subgroup of S, generated by the transpositions (ji+l) withj#i. If l<a,<a,<...<ai<n and l<b,<b,<...<bi<n are the respective positions occupied by the l’s of u and u E { 0, I};, one finds

u<;o iff a,db,,a,<b, ,..., aidbi. (9)

For the details of this construction for an arbitrary Coxeter group, the reader may consult [D] (particularly Lemma 3.6). A case-by-case treat- ment for the classical Weyl groups (including S,) is given in [PI, where the characterization of the Bruhat order implied by (8) and (9) is referred to as the tableau description.

The diagrams of the quotients < L and < i corresponding to S5 are given in Fig. 3. The remaining quotients ( < i, < “,) may be obtained by using the

00011

/ 00001 00 10’1

I 00010 /\ 01001 00110

I 00100 /\/ 10001 01010

I 01000 \/\ 10010 01100

I 10000 \/ 10100

\ 11600

FIGURE 3


0000 1

00011

/ OOlOl

/\ 01001 00110

/\/ 10001 01010

\/ 10010

\ 01100

\ 10100

\ 11000

FIGURE 4

fact that the change of alphabet 0 w 1 induces an anti-isomorphism <&+b <;I-‘.

To make the exponential/Bruhat analogy even more explicit, we may likewise use the notation < b (1 ,< i < n) for the restriction of < o to { 0, 1) y and observe that Theorem 3.1 implies

u<,v iff cp,(u) <b q,(v) for l<i<n.

The diagrams of the quotients CL and <i corresponding to S, appear in Fig. 4. We remark that the simplest way to prove that < s = < E for S4 is to simply verify that < B = -C & for i = 1, 2, 3.

Although explicit diagrams of the Bruhat and exponential orders for S, are best left to the imagination, the discrepancies between cB and cE are evident in Figs. 3 and 4. There are four new relations in the exponential quotient orders that are not present in the Bruhat quotients: 01100 ~$10010, 01100 -CL 10001, and their duals, 01101 -c; 10011, 01110 -c: 10011. Hence, the new relations in (S,, <J that do not occur in (S,, < B) must involve (at least) one of these four quotient relations. By an elementary (but tedious) analysis, it can be shown that there are 104 exponential relations not in the Bruhat order. Moreover, these new relations are all transitive consequences of the Bruhat relations, together with the 7 relations,

43251~~34521 inv=7


42153~~24513 inv=5

32154 <E 23451 32154 <E 23514

inv = 4

32145 cE 23415 inv = 3.

Note that each of these new relations involve pairs of the same Bruhat rank. Therefore, we may conclude that u GE v implies inv(u) < inv(v) for all U, v E Ss. This suggests the conjecture u GE v =inv(u) < inv(v) for all U, v E S,, a possibility mentioned in [Br, Conjecture B]. However, we have 543216 cE 345612, whereas inv(543216) = 10 and inv(345612) = 8, so this conjecture fails for n > 6.

Finally, we remark that if the domain r, for the exponential order is replaced by a domain bounded sufficiently far from e, then the exponential order changes only slightly. In this case, Theorems 2.2 and 3.1 remain valid if the lexicographic order < L is substituted for the quotient order < o; the proofs remain essentially unchanged.

ACKNOWLEDGMENTS

I would like to thank Alex Andretta for valuable discussions. Also, I would like to thank Jerry Griggs for correcting an error in a previous version of this paper.

REFERENCES

[Bj] A. BJ~RNER, Orderings of Coxeter groups, in “Combinatorics and Algebra,” Contem- porary Math. Vol. 34, pp. 175-195, Amer. Math. Sot., Providence, RI, 1984.

[SW] A. BJ~RNER AND M. WACHS, Bruhat order of Coxeter groups and shellability, Adv. in Math. 43 (1982), 87-100.

[Br] B. W. BRUNSON, The partial order of iterated exponentials, Amer. Math. Monthly 93 (1986), 779-786.

[D] V. V. DEODHAR, Some characterizations of Bruhat ordering on a Coxeter group and determination of the relative Mobius function, Invent Math. 39 (1977), 187-198.

[P] R. A. PROCTOR, Classical Bruhat orders and lexicographic shellability, J. Algebra 77 (1982), 104126.

Documents

The Bruhat order and iterated exponentials