Analogue of the Bruhat–Chevalley Order for Reductive Monoids

196, 339]368 Ž1997. JA977111

JOURNAL OF ALGEBRA ARTICLE NO.

Analogue of the Bruhat]Chevalley Order for Reductive Monoids Edwin A. Pennell Department of Mathematics, Bluefield College, Bluefield, Virginia 24605

and Mohan S. Putcha Department of Mathematics, North Carolina State Uni¨ ersity, Raleigh, North Carolina 27695

and Lex E. Renner Department of Mathematics, The Uni¨ ersity of Western Ontario, London, Ontario, N6A 5B7, Canada Communicated by Georgia Benkart Received September 26, 1995

INTRODUCTION The purpose of this paper is to describe the Adherence Order of B = B-orbits on a reductive algebraic monoid. By the results of w10x there is already a perfect analogue of the Bruhat decomposition for reductive monoids. Indeed, by w10, Corollary 5.8x, if M is reductive with unit group G and maximal torus T : G with Borel subgroup B = T, then the set of two sided B-orbits, B R MrB is canonically identified with a certain finite inverse monoid R. In fact, R s NG ŽT .rT, the orbit monoid of the Zariski closure of NG ŽT . in M. By definition, for s , t g R we define the Adherence Order by sFt if Bs B : Bt B, where Bt B denotes the Zariski closure in M of Bt B. 339 0021-8693r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.

340

PENNELL, PUTCHA, AND RENNER

In the case of reductive groups, a similar order relation was first studied by Chevalley. Since then Tits w13x developed the notion of a ‘‘BN pair’’ or ‘‘Tits system’’ which provided the abstract framework for much subsequent work in group theory Žincluding complete homogeneous spaces and Schubert varieties w2x.. Furthermore, the much studied KL-polynomials are quantified in terms of the Bruhat]Chevalley order on W w3x. For reductive monoids one would like to describe the Adherence Order on R in terms of the Bruhat order on W along with some other necessary invariant. This other invariant turns out to be the I-order. For n = n matrices, A G B in the I-order, if rankŽ A. G rankŽ B .. Furthermore, each element s of R can be expressed uniquely in standard form s s xeyy1 , where e g L the cross section lattice, and x, y g W are appropriately restricted Žsee Corollary 1.5 for details.. Our main result is the following: THEOREM. Let s s xeyy1 and t s sfty1 be in standard form. Then the following are equi¨ alent: Ža. Žb.

sFt e F f and there exists w g W Ž f .We such that x F sw and tw F y.

Here e F f means simply that ef s fe s e. This is the I-order condition mentioned above. As usual, W Ž f . s  w g W < wf s fw4 and We s  w g W < we s ew s e4 . In Section 2 we analyze further the Adherence Order. To do this, we introduce the j order on R. Indeed, we define

s Fj t

if Bs B F Bt B.

Clearly, s Fj t implies s F t . The main result of Section 2 is essentially the following theorem. THEOREM. Suppose s F t . Then there exists u 0 , u 1 , . . . , um g R such that s s u 0 F u 1 F u 2 F ??? F um s t , and for all i F m either Ža. u i Fj u iq1 or else Žb. u i and u iq1 differ by a Bruhat interchange Ž see Definition 2.9.. This determines the Adherence Order in terms of the two simpler relations. The relation of Ža. is not encountered in group theory but is easily analysed via Theorem 2.7. The relation of Žb. is described in more detail in Definition 2.9 and Theorem 2.11. ŽFor MnŽ k . it implies that u i and u iq1 have the same nonzero rows and columns.. It is similar to the situation encountered in the study of the Bruhat]Chevalley order on a Coxeter group w4, Chap. 5x. In Section 3 we apply the general results of Sections 1 and 2 to the monoid MnŽ k . of n = n matrices. In this example, R s Rn can be identified with the finite monoid of zero-one matrices with at most one

REDUCTIVE MONOIDS

341

nonzero entry in each row or column. These are sometimes referred to as partial permutation matrices. We use the results of Section 1 or 2 to find a purely combinatorial description of the Adherence Order on Rn . In Section 4 we discuss a natural length function on R. Here we continue our policy of determining all ‘‘monoid quantities’’ in terms of ‘‘L-quantities’’ and ‘‘W-quantities.’’ For example, if s s xeyy1 g R is in standard form we define the length of s by lŽ s . s lŽ x . q lŽ e. y lŽ y . , where l Ž x ., l Ž y . are the lengths of x and y as elements of the Coxeter group ŽW, S . Ž l Ž e . is defined in Section 4.. We show in Section 2 that length is subadditive, namely l Ž st . F l Ž s . q l Ž t . . In the final section of the paper we pursue some further properties of the Adherence Order. While the results of this section could benefit from some further refinements, we include them for future reference.

1. THE ADHERENCE ORDER Let M be a reductive algebraic monoid with maximal torus T and Borel subgroup B = T. EŽT . s  e g T < e 2 s e4 . Let L s  e g EŽT .< Be : eB4 be a cross section lattice, and let R s NG Ž T . rT. Recall from w10, Corollary 5.8x that R can be canonically identified with the set of two sided B-orbits on M. For s , t g R we define

sFt

Ž 1.

if Bs B : Bt B ŽZariski closure in M .. It is easy to verify that F determines a partial order on R. The main result of this section describes this order relation in terms of Ži. the I-order on M Žii. the Adherence Order on W. The I-order Žon M . has been determined explicitly in much detail by the second and third named authors for a large class of reductive monoids w8x. The Adherence Order on W is the much studied Bruhat]Chevalley order. Before stating our main result we recall some relevant background information. Let G be a reductive group with T : B : G as usual. Let W be the Weyl group of T. By definition, if x, y g W, then x F y if and only if BxB : ByB.

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1.1. LEMMA. x F y if and only if By yB : By xB where By is the Borel subgroup opposite to B. Proof. Let w g W be the longest element. Then by w4, Sect. 5.9x, x F y iff wy F wx iff BwyB : BwxB iff wBwyB : wBwxB iff By yB : By xB, since Bys wBw. 1.2. LEMMA.

x F y iff xByy1 l ByB / B.

Proof. By w3, Corollary 1.2x, x F y iff By xB l ByB / B. But ByxB l ByB / B m ByxB l By / B m ByxB l ByBy / B m ByxByy1 l ByB / B m xByy1 l By B / B. 1.3. LEMMA.

For all x g W, ByxB : ByBx l xByB.

Proof. Write xBxy1 s UyUT where Uy: By and U : B. Then xBxy1 s UyŽ UT . : By B. So xB : By Bx. Thus, By xB : By Bx. Similarly, xy1 By x : By B. Thus, By xB : xBy B as well. For I : S let L I be the associated Levi factor and BL s L I l B. Let x g W. Then by w1, Proposition 2.3.3x x has minimal length l Ž x . in WI x m xy1 BL x : B.

Ž 2.

Now let M be a reductive monoid with unit group G and cross section lattice L. For e g L let W Ž e . s CW Ž e . We s  w g W we s e 4 eW Ž e . De s  x g W x has minimal length in xWe 4 D Ž e . s  x g W l Ž x . is minimal in W Ž e . x 4 L Ž e . s CG Ž e . ,

H Ž e . s eL Ž e .

B Ž e . s CB Ž e . By Ž e . s CBy Ž e . . If CW Ž e . s ² s < s g I : and x g De then Bex s B Ž e . x s exxy1 B Ž e . x : exB,

by Ž 2 . .

REDUCTIVE MONOIDS

343

Hence, BexB : exB : BexB and so BexB s exB

for x g De .

Ž 3.

Now let e, f g L with e F f. Then W Ž e . s We W Ž e . l W Ž f . .

Ž 4.

Clearly, R s WLW s

D WeD Ž e . .

Ž 5.

egL

Let x g W, e g L, y g DŽ e ., and assume x9 F x. Then by Lemma 1.3 and Ž3., Bx9eyB : Bx9BeyB : BxBeyB : BxBeyB s BxeyB. Thus x9ey F xey

if y g D Ž e . and x9 F x.

Ž 6.

Finally, let y g DŽ e . and assume x F yy1 . Then by Ž6., xey F yy1 ey. Hence xey g B

if y g D Ž e . and x g W , x F yy1 .

Ž 7.

1.4. THEOREM. Let e, f g L, x, s g W, y g DŽ e ., and t g DŽ f .. Then the following are equi¨ alent: Ža. xey F sft Žb. ef s e and there exist w g W Ž f .We , z g We such that wy1 t F y and x F swz in W. Proof. Let e, f, s, t, x, and y be as above and assume Žb. holds for w and z as indicated. Now by Ž7., ty1 wey g B. So sfwey s sftty1 wey g sftB : BsftB. So sfwey F sft. By assumption, w s w 1w 2 with w 1 g w Ž f . and w 2 g We . Thus, sfwey s sfw 1w 2 ey s sfw 1 ey s sw 1 ey s sw 1w 2 ey s swey. But by Ž6., xey F swzey F swey F sfwey F sft. Conversely, suppose xey F sft. Then clearly e F f and xey g BsftB. Hence, e g xy1 BsftByy1. Now, for w g W, let A w s xy1 Bs l BywB.

Ž 8.

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So xy1 Bs s Dw g W A w . Since this is a finite disjoint union of subvarieties, there exists a unique w g W such that A w : xy1 Bs is open and dense. So e g A w ftByy1 . Thus, e g eA w ftByy1 e : eA w ftByy1 e: eBy wBftByy1 e: eBy ewfBftByy1 e. Hence, ewf I e. So wfwy1 G e. But f G e by assumption. Thus, there exists ¨ g CW Ž e . such that ¨ f¨ y1 s wfwy1. But then ¨ g W Ž f . l W Ž e . / B, say w s ¨ c with c g W Ž f .. Hence, by Ž4., w s ¨ c g W Ž e .W Ž f . : We w W Ž e . l W Ž f .xW Ž f . : WeW Ž f .. Conclude that w s w 1w 2

for some w 1 g We , w 2 g W Ž f . .

Ž 9.

Since A w / 0, we see by Lemma 1.3 that B / xy1 Bs l BywB : xy1 Bs l By Bw. Thus xy1 Bswy1 l By B / B. So by Lemma 1.2 and Ž9. y1 x F swy1 s swy1 2 w1 .

Ž 10 .

By Ž10. e g By w 1w 2 BftByy1 s By w 1w 2 ftByy1 ,

by Ž 3 .

s By w 1 fw 2 tByy1 . For u g W, let Cu s w 2 tByy1 l ByuB. Then w 2 tByy1 s

D Cu .

ugW

Since this is a finite union we see that for some u g W, Cu : w 2 tByy1 is open and dense. Thus, e g By w 1 fCu . So e g eBy w 1 fCu e : eBy w 1 fCu e : eByy w 1 fBy wBe : eBy ew1 fBy uBe : eBy efBy uBe s eBy eBy uBe s eBy eueBe. Thus, eue I e. Hence u g W Ž e .. So in H Ž e ., e g eBy eueBe. By Lemma 1.1 applied to H Ž e ., eu F e in W Ž H Ž e ... Hence, eu s e. So u g We . Since

345

REDUCTIVE MONOIDS

Cu / B, we see from Lemma 1.3 that B / w 2 tByy1 l BywB : w 2 tByy1 l uByB. So uy1 w 2 tByy1 l ByB / B. Thus, by Lemma 1.2, uy1 w 2 t F y.

Ž 11 .

y1 y1 Ž . Let w w 1 g We . By Ž10. and Ž11. ˜ s wy1 2 u g W f We , z s u y1 y1 y1 y1 x F swy1 w 1 s swz, ˜ 2 w 1 s sw 2 uu

w ˜y1 t s uy1 w 2 t F y.

and

One can streamline the theorem somewhat as follows. It is easy to check that each element s g WeW, with e g L, can be written uniquely as

s s xeyy1 ,

x g De , y g D Ž e . .

We call this the standard form for s . EXAMPLE. Let M s M3 Ž k ., and let T be the group of diagonal invertible matrices, B the group of upper triangular invertible matrices. One checks that Ls

½ž

1 0 0

0 1 0

0 1 0 , 0 1 0

0 1 0

/ž

0 1 0 , 0 0 0

0 0 0

/ž

0 0 0 , 0 0 0

/ž

0 0 0

0 0 0

/5

.

So let 0 ss 0 1

1 0 0

1 0 0

0 0 0

ž

0 0 g R3 . 0

/

Then 0 ss 0 1

ž

1 0 0

0 1 0

/ž

0 1 0

/ž

1 0 0

0 1 0

0 0 s Ž 132 . e Ž 1 . . 1

/

One easily checks that this is a standard form since De s W and DŽ e . s  1, Ž23., Ž123.4 . 1.5. COROLLARY. Let s s xeyy1 and t s sfty1 be in standard form. Then the following are equi¨ alent: Ža. Žb.

s F t. e F f and there exists w g W Ž f .We such that x F sw and tw F y.

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PENNELL, PUTCHA, AND RENNER

Proof. This is straightforward. Remark 1. Let s s xeyy1 , t s sfty1 be in standard form. The characterization in Corollary 1.5Žb. is ‘‘best possible’’ in the sense that W Ž f .We cannot be replaced by W Ž f .. For example, let 1 ss 0 0

ž

0 0 0

0 0 0

/

and

1 ts 0 0

ž

0 0 0

0 1 . 0

/

Coincidently, the same example shows that s F t « u l Ž s . y l Ž e . F l Žt . y Ž . l f . On the other hand, if s F t and s g Wt W then l Ž s . F l Žt .. Remark 2. It is interesting to notice that  s g R < s F 14 s  s g R < s g B4 can be identified with a subset of Ž x, y . g W = W < x F y4 . Indeed, write s s xeyy1 in standard form. Then by Lemma 2.3 below x F y. One might hope for an interesting relationship with Kazhdan]Lusztig polynomials. Indeed, the KL-polynomials are indexed by ordered pairs in W. There is already some connection between KL-polynomials and the Bruhat decomposition established by the second named author w7x.

2. THE j-ORDER ON R 2.1. DEFINITION.

Let r, s g R. We say r Fj s if BrB : BsB.

One checks easily that F j is a partial order on R. The major purpose of this section is to find a description of F j entirely within R Žsee Theorem 2.7 below.. Along the way, we shall need to establish several other important results about F and F j . 2.2. DEFINITION.

Define Rq: R as Rqs  r g R < BrB : B4 .

One checks that this is well defined. For MnŽ k ., Rq is the set of upper triangular partial permutation matrices. 2.3. LEMMA. only if x F y.

Let r s xeyy1 be in standard form. Then r g Rq if and

Proof. If r g Rq then r F 1. Applying Corollary 1.5 we find that there exists w g W such that x F w F y. In particular, x F y. Conversely, if x F y, then plainly the criteria of Corollary 1.5Žb. are satisfied for s s r and t s 1. It was established in w10x that if s g S, the simple reflections, and r g R, then sBr : BrB j BsrB

REDUCTIVE MONOIDS

347

and rBs : BrB j BrsB. We use this to help prove Theorems 2.4, 2.5, and 2.6 below. 2.4. THEOREM. Dy F x ByrB.

If r g R and x g W then rBx : Dy F x BryB and xBr :

Proof. We proceed by induction on the length of x. By the results of w13x, we know that both inclusions hold if l Ž x . s 1, since s s  x g W < l Ž x . s 14 . Assume the results hold if l Ž x . - k, and suppose l Ž x . s k. Then there exists s1 , s2 , . . . , sk g S such that x s s1 s2 ??? sk . By induction we know that rBx s Ž rBs1 ??? sky1 . sk : ŽD y F x BryB . sk , where x9 s s1 ??? sky 1 s xsk . But BryBsk : BryB j Brysk B from w10, Proposition 5.3x. Thus rBx : Dy F x 9 Ž BryB j Brysk B .. Hence rBx : Dy F x BryB. The other case is similar. 2.5. THEOREM.

If r, r 1 g R then rBr1 : Dr 2 F r 1 Brr 2 B.

Proof. Write r 1 s xeyy1 in standard form. Then by Theorem 2.4 we have that rBr1 s rBxeyy1 : Dx 1 F x Brx 1 Beyy1 . Now let L s  x g G < xe s ex 4 s CG Ž e ., BL s B l L s CB Ž e .. Then Be s eBe s CB Ž e . e by w6, Theorem 6.16, Corollary 6.34x. Hence Brx 1 Beyy1 s Brx 1CB Ž e . eyy1 s Brx 1 eCB Ž e . yy1 . But, y g D Ž e . and so yC B Ž e . y y 1 : B. Thus, Brx 1 C B Ž e . s Brx 1 eyy1 yCB Ž e . yy1 : Brx 1 eyy1 B. But notice that x 1 F x and y g DŽ e ., so that r 2 s x 1 eyy1 F xeyy1 s r 1. Hence, rBr 1 : Dx 1 F x Brx 1 Beyy1 : Dx 1 F x Brx 1 eyy1 B : Dr 2 F r 1 Brr 2 B. 2.6. THEOREM.

If r, r 1 g R then r 1 Br : Dr 2 F r 1 Br 2 rB.

Proof. By w9, Theorem 8.2x, there exists an involution t : M ª M so that

t 2Ž x. s x t Ž xy . s t Ž y . t Ž x .

for all x g M, for all x, y g M,

t < T s id and

t Ž B . s By. Now let w g NG ŽT . represent the longest element of W Žso that wBwy1 s By. , and define u s intŽ w .(t . In particular, u Ž B . s B and u Ž xy . s u Ž y .u Ž x . for x, y g R.

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PENNELL, PUTCHA, AND RENNER

If r 1 , r g R, then by Theorem 2.5, rBr1 : Dr 2 F r 1 Brr 2 B. So also, u Ž r 1 . Bu Ž r . : Dr 2 F r Bu Ž r 2 .u Ž r . B. But r 2 F r 1 if and only if u Ž r 2 . F u Ž r 1 . since u Ž B . s B and u is a homeomorphism in the Zariski topology. So our conclusion follows by applying u directly to the inclusion u Ž r . Bu Ž r 1 . : Dr 2 F u Ž r 1 . Bu Ž r .u Ž r 2 . B of Theorem 2.5. 2.7. THEOREM. Bu B.

Let s , u g R. Then s g Rqu Rq if and only if Bs B :

Proof. If s g Rqu Rq then s s r 1 u r 2 where r 1 , r 2 g Rq. Thus, Bs B s Br1 u r 2 B : Br1u r 2 B : Bu B

since Br1 , r 2 B : B.

Conversely, if Bs B : Bu B then there exist r 1 , r 2 g Rq such that Bs B : Br1 Bu Br 2 B. On the other hand, by Theorem 2.6, r 1 Bu : Dr 3 F r 1 Br 3 u B, while by Theorem 2.5, r 3 u Br 2 : Dr 4 F r 2 Br 3 u r4 B. Hence Bs B : Br1 Bu Br 2 B : D r Fr Br 3 u r4 B, from which it follows that s s r 3 u r4 for some 3

1

r 4 Fr 2

r 3 F r 1 and r4 F r 2 . But r 1 , r 2 g Rq and so r 3 , r4 g Rq. Theorem 2.7 above gives us one of the major ingredients in the decomposing the Adherence Order on R. It turns out that any relation s F t can be obtained as a chain of elementary relations s s u 0 - u 1 - u 2 - ??? - un s t , where, for each i, either u i g Rqu iq1 Rq, or else u iq1 is obtained from u i by a ‘‘Bruhat Interchange’’ Žsee Definition 2.9 below.. The next theorem will get us closer to the major result of this section. 2.8. THEOREM. Let b g RefŽW . s Dw g W wSwy1 and x, y g W. Assume that x - x b . Then the following are equi¨ alent. Ža. Žb.

Bxeyy1 B : Bx b eyy1 B and xeyy1 / x b eyy1 b f W Ž e . s  x g W < xe s ex 4 .

Proof. Assume Žb.. So b g RefŽ w . RW Ž e .. Then x b can be written in reduced form as x b s s1 ??? sr where si g S for all i, s1 , . . . , sm g DŽ e . and smq 1 , . . . , sr g W Ž e .. Clearly, Ž x b . b s x and Ž x b . b - x b . Thus, by the strong exchange condition, there exists a unique i such that x s Ž x b . b s s1 ??? siy1 siq1 ??? sr . Hence b s Ž sr ??? siq1 . si Ž siq1 ??? sr .. Also, xe b xy1 s s1 ??? siy1 siq1 ??? sr esr ??? s1 .

349

REDUCTIVE MONOIDS

But now b g RefŽW . RW Ž e ., so we must have i F m. Expanding further, we obtain xe b xy1 s s1 ??? siy1 siq1 ??? sm esm ??? s1 with

s1 ??? siy1 siq1 ??? sm - s1 ??? sm

and s1 ??? sm g D Ž e . . But from Lemma 2.3 we obtain that xe b xy1 g Rq. Hence, Bxeyy1 B : Bxeyy1 B s B Ž xe b xy1 .Ž x b eyy1 . B : Bx b eyy1 B. Finally, if b f W Ž e . then b f We . Hence, e / b e and it follows that xeyy1 / x b eyy1 . Conversely, assume b g W Ž e .. Thus, xe b yy1 s x b eyy1. Now if xe b xy1 g B then e b s e s b e and so xeyy1 s x b eyy1. Hence, either xe b xy1 f B or else xeyy1 s x b eyy1. If xe b xy1 f B then Bxeyy1 B ­ Bx b eyy1 B. ŽFor if Bxeyy1 B : Bx b eyy1 B then by Theorem 2.7, there exists u, ¨ g Rq such that ux b eyy1 ¨ s xeyy1 . Replacing u and ¨ by uxexy1 and yeyy1 ¨ , if necessary, we can assume u g x exy1 CW Ž xexy1 . and ¨ g yeyy1 CW Ž yeyy1 .. But also, u, ¨ g Rq and hence u s xexy1 and ¨ s yeyy1 . But then xeyy1 s xe b yy1 . But this implies e s e b and so xe b xy1 g B, a contradiction.. We conclude that if b g W Ž e . then either xeyy1 s x b eyy1 or else Bxeyy1 B ­ Bx b eyy1 B. This completes the proof. 2.9. DEFINITION. Let xeyy1 and sety1 be in standard form, xeyy1 / sety1 . A Bruhat Interchange occurs between xeyy1 and sety1 if there exists a g RefŽW . l W Ž e . such that x F x a and x a eyy1 s sety1. Notice that for xeyy1 / x a eyy1 , x F x a if and only if xeyy1 F x a eyy1 . 2.10. EXAMPLE. Consider, in M3 Ž k ., 1 ss 0 0

ž

0 0 0

0 1 1 s 0 0 0

0 0 1

1 0 0

0 1 0

0 0 0

1 0 0

0 1 0

1 0 0

0 0 1

0 1 s xeyy1 0

1 0 0

0 0 1

/ ž /ž /ž / ž / ž /ž / ž /ž /

0 ts 1 0 s

0 1 0

1 0 0

0 0 0

0 1 0

s x a eyy1 .

1 0 0

0 0 1

0 1 0

1 0 0

0 0 1

0 0 0

0 1 0

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PENNELL, PUTCHA, AND RENNER

Here 1 es 0 0

ž

0 1 0

0 0 0

/

ž

and

0 1 0

1 0 0

0 0 g W Ž e. . 1

/

Clearly, 1 xs 0 0

0 1 0

ž

0 0 0 F 1 1 0

/ ž

1 0 0

0 0 s xa . 1

/

2.11. THEOREM. Let s s xeyy1 and u s sfty1 be in standard form. Then s F u if and only if there exist u 0 , u 1 , . . . , ur g R such that s s u 0 F u 1 F ??? F ur s u , and for each l, either u ly1 Fj u l or else u ly1 and u l differ by a Bruhat Interchange. Proof. Obviously, the condition is sufficient. So assume s F u . Then Bxeyy1 B : Bsfty1 B, and there exists w g W Ž f .We such that x F sw and tw F y. Write w s w 1w 2 with w 1 g W Ž f . and w 2 g We . using this fact, we obtain sfweyy1 s sfw 1w 2 eyy1 s sw 1 few 2 yy1 s sw 1 ew 2 yy1 s sweyy1 . Using this, we also obtain Bsweyy1 B : Bsfty1 tweyy1 B : Bsfty1 B,

since tweyy1 g Rq.

But also Bxeyy1 B : Bsweyy1 B, since x F sw. In any case, we obtain xeyy1 F sweyy1 Fj sfty1 . So we let ur s sfty1 and ury1 s sweyy1 . Then we only need to find the chain of u ’s from xeyy1 to sweyy1. Since x F sw we can find w4, Proposition 5.11x g 1 , . . . , g k g RefŽW . such that x - xg 1 - xg 1 g 2 - ??? - xg 1 ??? gr s sw and for every i g  1, 2, . . . , r 4 there exists d i g We such that xg 1 ??? g i d i g

351

REDUCTIVE MONOIDS

De . Since x g DŽ e ., x F xg 1 d 1 F xg 1 g 2 d 1 F ??? - xg 1 g 2 ??? gr dr . If xg 1 g 2 ??? gr dr s x then y1 sweyy1 s swdy1 s xg 1 ??? gr eyy1 s xeyy1 . r ey

In this case, we can just let u 0 s sweyy1 and u 1 s sfty1 and we are done. If xg 1 g 2 ??? gr / x then let

u i s xg 1 ??? g i d i eyy1 s x Ž g 1 ??? g iy1 . g i eyy1 . If g i g RefŽW . RW Ž e . then by Theorem 2.8, Bu iy1 B : Bu i B, while if g i g RefŽW . l W Ž e . then u iy1 and u i differ by a Bruhat Interchange. 2.12. EXAMPLE. Let M s M4Ž k .. Then 0 0 ss 0 0

0 0 0 0



1 0 0 0

0 0 1 0 F 0 0 0 0

0 0 1 0

0

0 0 0 0

1 0 s u. 0 0

0

We can illustrate Theorem 2.11 as 0 0 u0 s 0 0

0 0 0 0



1 0 0 0

0 1 F u1 s 0 0

0 0 0 0

1 0 0 0

0 0 0 0

0 1 0 0

0  0  0  0

0 0 F u2 s 0 0

1 0 0 0

0 0 0 0

0 0 F u3 s 1 0

0 0 0 0

0 0 1 0

0 0 0 0

1 0 . 0 0

One could invoke Theorem 2.8 to prove u 0 Fj u 1 Fj u 2 . However, we can see this directly Žusing Theorem 2.7. by observing

u0 s u1



1 0 0 0

0 0 0 0

0 1 0 0

0 0 0 1

0

and

1 0 u1 s 0 0



0 0 0 0

0 1 0 0

0 0 u . 0 2 1

0

Definition 2.9 easily yields that u 2 ¬ u 3 is a Bruhat Interchange. ŽThe Bruhat Interchange for MnŽ k . will be discussed systematically in Section 3..

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3. THE BRUHAT]CHEVALLEY ORDER ON MnŽ k . In this section we illustrate our theory for the reductive monoid MnŽ k . of n = n matrices. In this example, the symmetric inverse semigroup R s Rn can be identified with the set of n = n zero-one monomial matrices. Our approach in this section is to identify and interpret the main results of Section 2 Žespecially Theorem 2.11. as they apply to MnŽ k .. To help focus the reader, we consider the following question: Is there a purely combinatorial description for the Adherence Order on Rn? 3.1. THEOREM.

Let Ei j be the n = n matrix Ž a st . such that

a st s

½

Ž s, t . s Ž i , j . Ž s, t . / Ž i , j . .

1, 0,

Then Ei j Fj Ek m if and only if i F k and m F j. ŽŽ i, j . is ‘‘up’’ and ‘‘to the right’’ of Ž k, m... Proof. If Ei j Fj Ek m then there exists upper triangular matrices X and Y such that XEk m Y s Ei j . However, this implies that Ž XEk m Y .st s 0 if s ) k or t - m. Thus, i F k and m F j. Conversely, suppose i F k and m F j. Then X s Ei k and Y s Em j are both upper triangular. But one calculates XEk l Y s Ei j . Remark. It is easy to check that Ei j Fj Ek m if and only if Ei j F Ek m . This leads to a number of special properties in the case of M s MnŽ k .. We refer to Ei j as an elementary matrix. 3.2. THEOREM.

Let A, C g Rn and write s

As

Ý Al ls1 t

Cs

Ý Cl , ls1

where  A l 4 and  Cl 4 are elementary matrices. Define SA s  A1 , . . . , A s 4 and SC s  C1 , . . . , Ct 4 . Then the following are equi¨ alent: Ža. A -j C Žb. There exists an injection u : S A ª SC such that A l F u Ž A l . for all l s 1, 2, . . . , s.

REDUCTIVE MONOIDS

353

Proof. Assume A Fj C. Notice that s s rankŽ A. and t s rankŽ C .. Write A s w1 C s w3

Is 0

0 w2 0

It 0

0 w4 , 0

ž / ž /

where w 1 , w 2 , w 3 , w4 g Sn , the unit group of Rn . Let X s w 1wy1 3 Y s w4y1

ž

Is 0

0 w2 . 0

/

Then we have XCY s A. Furthermore, one can check that, for all k F t, XCk Y F Ck , and rank Ž XCk Y . F 1. Hence, for each k s 1, 2, . . . , t either or else

XCk Y g SA XCk Y s 0.

But for any A l g SA there exists Ck such that XCk Y s A l Žsince XCY s A.. So define u Ž A l . s Ck , where XCk Y s A l . Conversely, if there exists an injection u : SA ª SC such that A l Fj u Ž A l . for all l s 1, 2, . . . , s, then we can find elementary upper triangular matrices X l and Yl such that X l u Ž A l .Yl s A l . Let s

Xs

Ý Xl ls1

and s

Ys

Ý Yl . ls1

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PENNELL, PUTCHA, AND RENNER

Then X and Y are both upper triangular and Žone checks. XCY s A. Thus A Fj C. 3.3. EXAMPLE. 1 As 0 0

ž

0 0 0

0 1 Fj 0

/ ž

1 0 0

0 1 0

0 0 sC 1

/

since SA s  E11 , E234 , SC s  E11 , E22 , E334 , and u : S A ª SC defined by u Ž E11 . s E11 , u Ž E23 . s E22 satisfies the criterion of Theorem 3.2. To properly illustrate Theorem 2.11 in the situation of Rn it remains to describe concretely the notion of a Bruhat Interchange ŽDefinition 2.9.. We begin our description with the rank two case. 3.4. THEOREM. Let A, C g Rn be such that A s Ei j q Ek l and C s Ek j q Eil . Then A - C if and only if i - k, j - l or i ) k, j ) l. Ž Notice that A - if and only if a Bruhat Interchange has occurred Ž see Definition 2.9... Proof. Suppose i - k and j - l Žthe other case is similar.. If E s E 11 q E 22 then A s Ž iy1 i . ??? Ž 12 .Ž ky1 k . ??? Ž 34 .Ž 23 . E Ž 23 . ??? Ž ly1l .Ž12. ??? Ž jy1 j . where Ž ab. is the permutation matrix which interchanges the ath row and the bth row. This is the standard form for A. One checks that C s Ž iy1i . ??? Ž 12 . Ž ky1k . ??? Ž 34 . Ž 23 .

Ž 12. E Ž 23 . ??? Ž ly1l . Ž 12 . ??? Ž jy1 j . . But wŽ iy1i . ??? Ž23.x - wŽ iy1i . ??? Ž23.xŽ12.. Therefore A - C. Conversely, assume i - k and j ) l or i ) k and j - l. But then the same argument, as above, shows that C - A. 3.5. THEOREM. Let A s Ei1 j1 q ??? qEi s j s g R Ž where, as usual, Ei j is an elementary matrix . and suppose C s Ei1 j1 q ??? qEi ky 1 jky 1 q Ei l jk q Ei kq 1 jkq 1 q ??? qEi ly 1 jly 1 q Ei k jl q Ejlq 1 jlq 1 q ??? qEi s j s Ž so C is obtained from A by interchanging two nonzero rows .. Relabel the two interchanged matrices Ei j and Ek l . Then A - C if and only if i - k, j - l or i ) k, j ) l. Proof. If i - k and j - l, then by Theorem 3.4, Ž Ei i q Ek k . A Ž Eii q Ek k .C. A little more calculation then shows that A - C. Conversely, if i ) k and j - l, or i - k and j ) l then the above argument shows that C - A. Notice that the situation above describes exactly the case where a Bruhat Interchange occurs between A and C.

REDUCTIVE MONOIDS

355

Combining Definition 2.9 and Theorems 2.11 and 3.5 we obtain the following description of the Adherence Order on Rn . 3.6. THEOREM ŽTheorem 2.11 for Rn .. Let s , t g Rn . Then s F t if and only if s s g 0 F g 1 F ??? F gr s t where, for each l, either Ža. g ly1 g Rnq g l Rnq , or else Žb. g l is obtained from g ly1 ¨ ia a Bruhat Interchange. To calculate in case Ža., we use Theorem 3.2, and in case Žb. we use Theorem 3.5. s Recall that for s g Rn we write Ss s  E1 , . . . , Es 4 if s s Ý is1 Ei and each Ei is an elementary matrix.

3.7. THEOREM. Suppose s F t and l Ž s . s l Žt . y 1. Ž So s F g F t implies s s g or g s t .. Then one of the following holds: Ža. < Ss < s < St < y 1 and Ss : St . Žb. < Ss < s < St < and < Ss l St < s < Ss < y 1 with Es - Et where Ss R Ž Ss l St . s  Es 4 and St R Ž Ss l St . s  Et 4 . Furthermore, s and t are either row equi¨ alent or column equi¨ alent. Žc. t is obtained from s ¨ ia a Bruhat Interchange. Proof. Assume Žc. is not the case, and rankŽ s . - rankŽt .. By Theorems 3.2 and 3.6 there exists an injection u : Ss ª St such that si F u Ž si . for each si g Ss . Define u Ž s . g Rn by Su Ž s . s u Ž Ss .. Then s F u Ž s . t . Thus s s u Ž s .. Therefore, Ža. holds. Now assume Žc. is not the case, and rankŽ s . s rankŽt .. By Theorem 3.6 we must have s s xt y for some x, y g Rq. Then s s xt y F t y F t . Hence, either s s t y or else t y s t . In the first case s s t y, and in the second case s s xt . Assume, without loss of generality that s s t y Ž s and t have the same nonzero rows.. Now by Theorem 3.2, the map u : Ss ª St has si F u Ž si . for all si g Ss . But by Theorem 3.1, this means that if si s Eu ¨ and u Ž si . s Er s then r G u and s F ¨ Ž Eu ¨ is ‘‘up’’ an ‘‘to the right’’ of Er s .. But s and t have the same nonzero rows. Since u is row increasing it follows easily that u is row preserving. So write Ss s  Ei1 , . . . , Ei s 4 and St s  Ej1 , . . . , Ejs 4 , where Ei k and Ejk are nonzero in the same row Žso that u Ž Ei k . s Ejk .. Now let Ei k g Ss be the elementary matrix with the maximum column

356

PENNELL, PUTCHA, AND RENNER

value. Define t 9 g Rn via St 9 s Ž St R  Ejk 94. j  Ei k 4 where Ejk 9 is nonzero in the same row as Ei k Žso u Ž Ejk . s Ejk 9 .. Then t 9 s Ý E g St 9 E g Rn since u is column decreasing. Also s Fj t 9 Fj t since Ei k has larger column value than Ei k 9. If t 9 / t then we are done, since we obtain s s t 9, and so Žb. holds. If t 9 s t Ži.e., Ei k s Ejk 9 . we move on; i.e., define t Ž2. using the Ei s g Ss with the next largest column value, thereby substituting it for the corresponding Ej s9 g St . Again one checks that t Ž2. g Rn , and s F t Ž2. F t . If t Ž2. s t then define t Ž3., and so on. Eventually, one obtains t Ž r . / t . But then s s t Ž r . since s F t Ž r . - t and l Žt . s l Ž s . q 1. So t is obtained from s by ‘‘moving’’ one elementary matrix of s ‘‘to the left.’’ So Žb. holds. Theorem 3.7 allows us to give a combinatorial description of the Adherence Order on Rn . First we represent the elements of Rn by sequences of nonnegative integers. Let s g Rn . We associate to s a sequence Ž e 1 ,???, e n . where, for all i, 1 F i F n, e i is defined as

ei s

½

if s is zero in the ith column if w s x k i s 1.

0, k,

EXAMPLE.



0 0 1 0

0 0 0 0

0 0 0 1

0 1 l Ž 3042 . . 0 0

0

Notice that a sequence Ž e 1 , . . . , e n . can occur this way for some r g Rn if and only if 0 F e i F n for all i; and whenever e i s e j , either i s j or else e i s e j s 0. For convenience we simply write s s Ž e 1 , . . . , e n . g Rn . We can now interpret Theorem 3.7 combinatorially. 3.8. THEOREM. The Adherence Order on Rn is the smallest partial order generated by declaring

s s Ž d 1 , . . . , dn . - t s Ž e 1 , . . . , e n . if either Ža. d j s e j for j / i and d i - e i Ž s is obtained from t by replacing some e i by a smaller number . Žb. d k s e k if k / i or j, i - j, d i s e i , e i s d j and e i ) e j Ž s is obtained from t by interchanging e i and e j Ž i - j . where e i ) e j ..

REDUCTIVE MONOIDS

357

Proof. We simply interpret each case of Theorem 3.7. Case 3.7Ža.. This says that s is obtained from t by setting some nonzero entry in t Žas a matrix. to zero. This is included in Theorem 3.8Ža.. Case 3.7Žb.. First assume that s and t have the same nonzero rows. So, as we saw in the proof of Theorem 3.7, the matrix s is obtained from the matrix t by moving a ‘‘one’’ to the right. This is included in Theorem 3.8Žb.. If s and t have the same columns then s is obtained from t by moving a ‘‘one’’ upward. This is included in Theorem 3.8Ža.. Case 3.7Žc.. This is easily seen as the special case of Theorem 3.8Žb. where both e i and e j are nonzero. 3.9. EXAMPLE. Let r s Ž21403. and s s Ž35201. in R5 . Then r - s, since

Ž 21403. - Ž 31402. ,

by Theorem 3.8 Ž b .

- Ž 34102. ,

by Theorem 3.8 Ž b .

- Ž 35102. ,

by Theorem 3.8 Ž a .

- Ž 35201. ,

by Theorem 3.8 Ž b . .

4. SUBADDITIVITY OF LENGTH In this section we consider a natural length on R, extending the much studied length function on W, the Weyl group. In keeping with our point of view in this paper, we shall define this length function in terms of the length function on W and a notion of length for the elements of L. This appears to be a natural generalization of length function considered by Solomon w12x for M s MnŽFq ., where R can be identified with the symmetric inverse semigroup on n letters. There is also a topologicalrgeometric definition for this length function. Indeed, one can define l Ž x . s dim Ž BxB . y dim Ž Bn B . , where n g WxW is the unique element such that Bn s n B. This is discussed in some detail by the third named author in w11x. Our main result in this section proves that the length function is subadditive.

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PENNELL, PUTCHA, AND RENNER

Let W s ² S : be the Weyl group, where S : W is the set of simple involutions relative to B and T : B. Define Ref Ž W . s

D xSxy1 .

xgW

For I : S we have Ref Ž WI . s WI l Ref Ž W . . Let l Ž I . s Ref Ž W . RRef Ž WI . D Ž I . s  x g W x has minimal length in xWI 4 . Thus, l Ž I . s max  l Ž x . x g DI 4 . For I, J : S let l Ž I, J . s Ref Ž WI . RWJ . So l Ž I . s l Ž S, I . and l Ž I, J . s l Ž I, I l J . s max  l Ž x . x g WI l DJ 4 .

Ž 1.

Now let R be as in section one with unit group W and cross section lattice L. For e g L, recall that W Ž e . s  x g W xe s ex4 We s  x g W xe s e 4 eW Ž e . .

˜ Ž e . of W such that W˜ Ž e . is Notice that there exists a unique subgroup W generated by a subset of S and ˜Ž e. W Ž e . s We = W D Ž e . s DI , De s DJ ,

where W Ž e . s ² I : , where We s ² J : ,

Ž 2.

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REDUCTIVE MONOIDS

and lŽ e. s l Ž I . . If e, f g L, let l Ž f , e. s l Ž J , I . ,

where W Ž f . s ² J : and W Ž e . s ² I : .

If e, f g L, x g W, then exf g WfW

iff e G f and x g W Ž e . W Ž f .

fxe g WfW

iff e G f and x g W Ž f . W Ž e . .

Ž 3.

If e, f g L then

˜Ž f . : W Ž e. . eGf«W

Ž 4.

e G f « W Ž f . s W Ž e, f . Wf ,

Ž 5.

Thus, by Ž2. and Ž4.,

where W Ž e, f . s W Ž e . l W Ž f .. Recall from the end of Section 1 that if s g WeW, e g L, then

s s xeyy1

for some x g De and y g D Ž e . .

Furthermore, x and y are unique Žby an easy counting argument.. 4.1. DEFINITION.

Let s s xeyy1 be in standard form. Define lŽ s . s lŽ x . q lŽ e. y lŽ y . ,

where l Ž x ., l Ž y . are the lengths of x and y as elements of W, and l Ž e . is as above. It is easy to see that l Ž z s . F l Ž z . q l Ž s . for any z g W. However, much more is true. The main result of this section ŽTheorem 4.5 below. says that for any s , t g R, l Ž st . F l Ž s . q l Žt .. For example, let 0 ss 0 0

ž

0 0 1

0 1 0

/

0 ts 1 0

ž

and

so that 0 st s 0 1

ž

0 0 0

0 0 . 0

/

1 0 0

0 0 , 0

/

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PENNELL, PUTCHA, AND RENNER

Then 0 ss 0 1

0 1 0

1 0 0

0 ts 1 0

1 0 0

0 0 1

ž ž

/ž /ž

1 0 0

0 1 0

0 0 0

1 0 0

0 1 0

0 0 0

/ž /ž

0 0 1

1 0 0

0 1 s xeyy1 , 0

1 0 0

0 1 0

0 0 s ue¨ y1 1

/ /

and 0 st s 0 1

ž

1 0 0

0 1 0

/ž

1 0 0

0 0 0

0 0 0

/ž

1 0 0

0 1 0

0 0 s rfsy1 1

/

determine the standard form of s , t , and st . Thus, by Definition 4.1 l Ž s . s l Ž x . q l Ž e. y l Ž y . s 3 q 2 y 2 s 3 l Ž t . s l Ž u. q l Ž e . y l Ž ¨ . s 1 q 2 y 0 s 3 l Ž st . s l Ž r . q l Ž f . y l Ž s . s 2 q 2 y 0 s 4. So l Ž st . F l Ž s . q l Žt . as required. Let e, f, h g L be such that eWf l WhW / B. Then

4.2. LEMMA.

l Ž h . F l Ž e . q l Ž f . y l Ž h, e . y l Ž h, t . . Proof. Let x g W be such that exf s ahby1 is in standard form. Then eah g WhW. So by Ž3. and Ž5. above a g W Ž e .W Ž h. s W Ž e .Wh . Since a g D h , we see that a g W Ž e .. Also, since hby1 f g WhW, we see by Ž3. that by1 g W Ž h.W Ž f .. Finally, since b g DŽ h., we see that b g W Ž f .. We conclude that eay1 xbf s ay1 exfb s h. Thus, without loss of generality we may assume that exf s h. So we obtain hxh s x and, by Ž3., x s W Ž h.. Since xy1 exf is an idempotent we obtain that xy1 exf s h. Thus, W Ž xy1 ex . l W Ž f . : W Ž h., and it follows that xy1 Ref Ž W Ž e . . x l Ref Ž W Ž f . . : Ref Ž W Ž h . . . Thus, in RefŽW ., c

c

Ref Ž W Ž h . . : xy1 Ref Ž W Ž e . . x j Ref Ž W Ž f . .

c

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REDUCTIVE MONOIDS

and so c

c

c

Ref Ž W Ž h . . : xy1 Ref Ž W Ž e . . x RRef Ž W Ž h . . l xy1 Ref Ž W Ž e . . x c

j Ref Ž W Ž f . . RRef Ž W Ž h . . l Ref Ž W Ž f . .

c

.

But Ref Ž W Ž h . .

c

s l Ž h.

c

xy1 Ref Ž W Ž e . . x s Ref Ž W Ž e . .

c

s l Ž e.

and Ref Ž W Ž h . . Rxy1 Ref Ž W Ž e . . x s xy1 Ref Ž W Ž h . . RRef Ž W Ž e . . x s l Ž h, e . since x g W Ž h.. Furthermore, Ref Ž W Ž f . .

c

s lŽ f .

Ref Ž W Ž h . . RRef Ž W Ž f . . s l Ž h, f . . Thus, we conclude that l Ž h. F w l Ž e . y l Ž h, e .x q w l Ž f . y l Ž h, f .x. 4.3. COROLLARY. Let e, f, h g L and x g DŽ e . be such that exy1 f s ahby1 is in standard form. Then l Ž a . q l Ž h . y l Ž b . F l Ž e . q l Ž f . y l Ž x . y l Ž h, f . . Proof. Now a g D h , b g DŽ h., and eah, hby1 f g WhW. Thus Ž3. and Ž5., a g W Ž e . and b g W Ž f .. Hence eay1 xy1 bf s ay1 exy1 fb s h. So hay1 xy1 bh s heay1 xy1 bfh s h. But then by Ž3., ay1 xy1 b g W Ž h., and so ay1 xy1 bh s h. Hence, ay1 xb g Wh . Let a s ay1 xy1 b g Wh . So ay1 xy1 s a by1 .

Ž 6.

Since a g W Ž e ., x g DŽ e ., a g Wh , and b g DŽ h., l Ž a . q l Ž x . s l Ž ay1 xy1 . s l Ž a by1 . s l Ž a . q l Ž b . .

Ž 7.

But a s a 1 a 2 with a 1 g Wh l W Ž e ., a 2 g Wh l DŽ e .y1 , and l Ž a . s l Ž a 1 . q l Ž a 2 .. So by Ž6., ay1 ay1 xy1 s a 2 by1. Then l Ž ay1 1 a. s l Ž a 1 . q l Ž a.

since a 1 g Wh , a g D h

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PENNELL, PUTCHA, AND RENNER

and y1 y1 y1 l Ž ay1 x s l Ž ay1 . q lŽ x. 1 a 1 a

s l Ž a 1 . q l Ž a. q l Ž x .

y1 since ay1 g W Ž e . and 1 a

x g DŽ e. . Furthermore, l Ž a 2 by1 . s l Ž a 2 . q l Ž b .

since b g D Ž h . .

Thus, l Ž a 1 . q l Ž a. q l Ž x . s l Ž a 2 . q l Ž b . . But l Ž a . s l Ž a 1 . q l Ž a 2 ., so by Ž7., l Ž a. q l Ž x . s l Ž a 1 . q l Ž a 2 . q l Ž b . . Conclude that l Ž a 1 . s 0, and so a s a 2 g Wh l DŽ e . : W Ž h. l DŽ e .. By Ž1., l Ž a . F l Ž h, e .. Hence, l Ž a. q l Ž x . F l Ž h, e . q l Ž b .. Combining this with the lemma we obtain l Ž a. q l Ž h. y l Ž b . F l Ž e . q l Ž f . y l Ž x . y l Ž h, f .. 4.4. COROLLARY. Let e, f, h g L, l Ž exy1 fty1 . F l Ž exy1 . q l Ž fty1 ..

x g D Ž e .,

t g D Ž f .. Then

Proof. Let exy1 f s ahby1 , h g L, be in standard form. Then as in Corollary 4.3, a g Dh l W Ž e . , b g D Ž h . l W Ž f .

Ž 8.

l Ž a . q l Ž h . y l Ž b . F l Ž e . q l Ž f . y l Ž x . y l Ž h, f . .

Ž 9.

and Let by1 ty1 s a ¨ y1 ,

a g W Ž f . , ¨ g D Ž h. .

Ž 10 .

By Ž8. and Ž10. l Ž b . q l Ž t . s l Ž by1 ty1 . s l Ž a ¨ y1 . s l Ž a . q l Ž ¨ . . .y1

Ž 11 .

But a s a 1 a 2 with a 1 l Ž f . l W Ž h. and a 2 g DŽ f l W Ž h.. So l Ž a . y1 y1 y1 Ž . s l Ž a 1 . q l Ž a 2 .. By Ž10., ay1 b t s a ¨ and 8 , lŽ a 1 . q lŽ b. s 1 2 y1 . y1 y1 Ž . Ž . l Ž ay1 b . Since a b g W h and t g D h , we obtain 1 1 y1 l Ž a 1 . q l Ž b . q l Ž t . s l Ž ay1 . q lŽ t. 1 b y1 y1 s l Ž ay1 t . 1 b

s l Ž a 2¨ y1 . s l Ž a2 . q l Ž ¨ . .

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REDUCTIVE MONOIDS

But by Ž11., l Ž b . q l Ž t . s l Ž a 1 . q l Ž a 2 . q l Ž ¨ .. Hence l Ž a 1 . s 0 and so a s a 2 g DŽ f .y1 l W Ž h.. By Ž1. l Ž a . F l Ž h, f . .

Ž 12 .

Since f G h it follows from Ž5. that

a g W Ž h . l D Ž f . : Wh l D Ž f . . Hence, by Ž10. exy1 fty1 s ahby1 ty1 s ah a ¨ y1 s ah¨ y1 . Finally, by Ž9., Ž11., and Ž12. l Ž exy1 fty1 . s l Ž a . q l Ž h . y l Ž b . y l Ž t . q l Ž a . F l Ž e . q l Ž f . y l Ž x . y l Ž h, f . y l Ž t . q l Ž a . , by Corollary 4.3 F l Ž e. q l Ž f . y l Ž x . y l Ž t . s l Ž exy1 . q l Ž fty1 . . 4.5. THEOREM.

Let s 1 , s 2 g R. Then l Ž s1 s 2 . F l Ž s1 . q l Ž s2 . .

Proof. Write s 1 s xeyy1 and s 2 s ¨ fty1 in standard form with e, f g L. Let yy1 ¨ s wyy1 where w g W Ž e . and y 1 g DŽ e .. Since y g DŽ e . and 1 y1 w g W Ž e . we obtain that wy1 yy1 s yy1 and l Ž wy1 yy1 . s l Ž w . q l Ž y .. 1 ¨ Hence, l Ž w . q l Ž y . s l Ž yy1 ¨ y1 . F l Ž y 1 . q l Ž ¨ . .

Ž 13 .

Finally, l Ž s 1 s 2 . q l Ž xeyy1 ¨ fty1 . y1 s l Ž xewyy1 . 1 ft y1 s l Ž xweyy1 . 1 ft y1 F l Ž xw . q l Ž eyy1 . 1 ft

F l Ž xw . q l Ž e . y l Ž y 1 . q l Ž f . y l Ž t . ,

by Corollary 4.4

F l Ž x . q l Ž w . q l Ž e . y l Ž y1 . q l Ž f . y l Ž t . F lŽ x . q lŽ ¨ . y lŽ y . q lŽ e. q lŽ f . y lŽ t . , s l Ž s1 . q l Ž s 2 . ,

by definition.

by Ž 13 .

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Remark. The question of subadditivity of length was raised by L. Solomon. It turns out that the issue of integral structure constants for monoid Hecke algebras hinges on Theorem 4.5. See the proof of w12, Theorem 4.12x for an illustration of this issue in the special case M s MnŽFq .. 5. RELATED RESULTS In this section we complete our current study by establishing a number of supplementary results concerning the Adherence Order. We focus on the relationship between the Adherence Order and the H-relation on R. It is clear to the authors that there are an intriguing number of unchartered possibilities that could be pursued here. Let M be a reductive monoid with maximal torus T. If s , t g R we say that s and t are H-related Žin R . if s R s t R and Rs s Rt . We write s Ht . Similarly, elements x and y of M can be H-related. We denote the H-class in M of x g M by H x and the H-class of x g R in R by Hx . So H x s  y g M < yM s xM and My s Mx4 . Two elements x and y of M are I-related Ž x I y . is MxM s MyM. We write x GI y if MyM : MxM. For n = n matrices, x GI y in the I-order if rankŽ x . G rankŽ y .. Let e, f g EŽT .. 5.1. LEMMA. Suppose x g eMf l eG. Then there exists e9 g EŽT . such that e9 g Cl W Ž e ., e9 F f, and xe9 g eG. Proof. Write x s eg and let b1 , b 2 g B be such that g s b1wb 2 . Here B is a Borel subgroup containing T such that B :  h g G < eh s ehe4 . Thus eb1 s eb1 e s ce for some c g CB Ž e .. Hence, eg s cewb 2 s cwewy1 b 2 s ge9b, where e9 s wy1 ew and b s b 2 . So xe9 s ege9 s ye9be9 g Ge9 since e9be9 g Be9. But also eg s x s xf. So if ege9 g Ge9 then f 9 G e9. Otherwise, f h e9 and so fe9 - e9 which implies that xfe9 - e9 in the I-order on M. But this is absurd since xfe9 s ege9 g Ge9. 5.2. COROLLARY. Suppose eBf l eG / B. Then there exists e9 g EŽT . such that e9 I e, e9 F f, and eBe9 l eG / B. Proof. Let x g eBf l eG and choose e9 as in Lemma 5.1. Then xe9 g eBe9 l eG and the conclusion follows. For s g R let Hs denote the H-class of s in M. 5.3. COROLLARY. Let r, s g R and let e, f g EŽT . be such that e R s s R and R s s R f. Suppose eBrBf l Hs / B. Then there exist e9, f 9 g EŽT . with e9 I f 9 I s such that eBe9rf 9Bf l Hs / B. In particular, e9r s rf 9 I s and if we write r s e0 s s s f 0 with s g W then e9 F e0 and f 9 F f 0 .

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Proof. Since eBrBf l Hs / B it follows easily that eBe0 l eG / B and f 0 Bf l Gf / B. So by Corollary 5.2 and its left-right dual we obtain e9 F e0 a nd f 9 F f 0 such that eBe9 l eG / B and f 9Bf l Gf / B. If we let e1 He 2 denote the H-class of e1 s where sy1 e1 s s e2 then it is easy to see that e1 He 2 e 2 He 3 :e1 He 3 . But we have B / eBe9r le H f 9 Žsince e9r s rf 9. and B / f 9Bf lf 9 H f . Thus, recalling that Hs se H f , we obtain eBe9rf 9Bf l Hs / B. We recall now some results from w11x. Define the set of order preser¨ ing elements of R by O s  r g R rBr* : Brr* 4 . Then by w11, Sect. 2x O : R is an inverse subsemigroup, EŽ R . : O , and
for each H-class of R .

For each s g R we can write uniquely s s sq s0 sy , where sq, syg O , sq R s s R, R sys R s, s0 H n , and n I s is the unique such element that satisfies Bn s n B. To illustrate these notions, consider our familiar example of Rn . One can easily calculate that, in this example, O s On s  s g Rn if si j , s k l / 0 and i - k, then j - l 4 ; i.e., for nonzero entries, the column value is an increasing function of the row value, e.g.,



1 0 0 0

0 0 1 0

0 0 0 1

0 0 g O4 , 0 0

0 

0 0 1 0

1 0 0 0

0 0 0 1

0 0 f O4 . 0 0

0

It is interesting to notice that for each s g Rn , there exists a unique s g On so that s be obtained form s via a sequence of Bruhat interchanges. On is an inverse semigroup because it is closed under matrix multiplication and transpose.

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Another easy calculation yields that

¡1

 n g Rn

Bn s n B 4 s

~

 

0 1 .. .

0 0 .. .

??? ??? ???

0 0

0 0

??? ???

0 0 .. .

0 0 .. .

??? ??? .. .

0 0 .. .

0 0

0 0

??? ???

0 0

0 .. .

¢ 00

0 0 0 0 .. . . , .. 0 0 1 0

1 0 .. .

0 1 .. .

??? ??? .. .

0 0 .. . ,???,

0 0

0 0

??? ???

1 0

1 0 0 0 .. . . , .. 0 0 0 0

0 0 .. .

??? ??? .. .

0 0 .. .

0 0 .. .

0 0

??? ???

0 0

0 0

0 0

0 0

¦

¥.

§

The decomposition s s sq s0 sy

Ž for any s g Rn .

can be thought of as follows: sq indicates the nonzero rows of s. sy indicates the nonzero columns of s. s0 indicates the nonzero columns as a function of the nonzero rows. One should think of s0 as the ‘‘order reversing’’ part of s. For example, s0 s n if and only if s g O . As an illustration, consider the example 0 1 0 1 ss 0 0 0 s 0 1 0 0 0 s sq s0 sy .

ž

/ ž

0 0 1

0 0 0

/ž

0 0 0

0 1 0

1 0 0

/ž

0 1 0

0 0 1

0 0 0

/

As promised, sq says ‘‘rows one and three are involved in s,’’ sy says ‘‘columns one and two are involved in s.’’ s0 represents the function R1 ¬ C2 R 3 ¬ C1 . See w11, Sect. 3x for more details concerning O and the q0 y decomposition. 5.4. DEFINITION. For s g R let s g O denote the unique element of O l Hs . So if s s sq s0 sy as above then s s sq n sy. For the elements of Rn , it is easy to find s from s without finding sq, s0 , and sy. s is the unique element of On with the same nonzero rows and

367

REDUCTIVE MONOIDS

columns as s. For example, if 0 1 ss 0 0



1 0 0 0

0 0 0 0

0 0 1 0

0

1 0 then s s 0 0



0 1 0 0

0 0 0 0

0 0 . 1 0

0

5.5. COROLLARY. The following are equi¨ alent for r, s g R with e R s s R and R s s R f. Ža. eBrBf l Hs / B Žb. r G s Žc. r G s. Proof. By w11, Corollary 4.4x, Hs : Dt H s BtB s BHs B, yet by continuity of multiplication eBrBf : BrB since e, f g T and TBrBT : BrB. Hence, H s such that r G t. On the other hand eBrBf l Hs / B iff there exists tH s g Hs is the unique smallest element in Hs . Thus, Ža. and Žb. are equivalent. But also r G r. So Žc. implies Žb.. So we may assume r G s. By Corollary 5.3 there exist e9, f 9 g R, I-equivalent to s, such that r G e9rf 9 and e9rf 9 G s. Thus, r G e9rf 9 s e9rf 9.

Ž 1.

H e9rf 9 and e9rf 9, e9rf 9g u . The equality here results from the fact that e9rf 9H But also e9rf 9 I s Žand e9rf 9 G s .. So by w11, Proposition 3.11x, e9rf 9 G s implies that Ž e9rf 9. "G s ". Thus e9rf 9 G s

Ž 2.

by w11, Proposition 3.2x since Že9r 9 f 9. 0 s Ž s0 .. Putting things together we obtain r G s « r G e9rf 9 G s « r G e9rf 9

Ž by Ž 1 . . .

But e9rf 9G s, by Ž2.. Remarks. Ž1. From the above proof we see that for r g R, s g O , and r G s there exists t I s such that t G s,

and

t s e9rf 9, where e9 R s t R and R t s R f 9.

This should be useful in getting a better picture of the Adherence Order generally. Ž2. Assume eBrBf l Hs / B where e R s s R and R s s R f. From w11, Corollary 4.4x we know that Hs : Dt H s BtB. Thus, there exists a

368

PENNELL, PUTCHA, AND RENNER

unique t g Hs such that eBrBf l BtB : eBrBf is dense. It would be nice to describe this t somehow. In any case we obtain the following corollary. 5.6. COROLLARY. r G t.

H s such that If r G s then there exists a maximum tH

5.7. EXAMPLE. We can use Theorem 3.8 to illustrate Corollary 5.6. Recall from Section 3 that the elements of Rn can be thought of as sequences of nonnegative integers r s Ž e 1 , . . . , e n .. So let n s 5, and consider r s Ž24105., s s Ž31020. g R5 . Then s s Ž12030. and one can easily check, using Theorem 3.8, that r G s. Again using 3.8, we see that r G Ž23010. s t. Furthermore, t is maximum in the Adherence Order with H s and r G t. the properties tH

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