BN-Pairs and Boolean Hecke Algebras

Todd Trimble

April 26, 2005

These are notes on a reworking of the theory of BN-pairs (as 
given in Brown's book) through a systematic study of the 
interplay between Boolean group algebras and Boolean Hecke 
algebras.  One benefit of this approach might be a basis for 
comparing various notions of "model" under development 
(e.g., buildings in the sense of Brown or Dolan, as compared 
to Boolean or Heyting Hecke algebras).  Another might be to 
cast the Brown/Tits theory of BN-pairs in more congenial (or 
less eye-glazing) terms. 

First a small comment which relates to our last discussion: 
while it's true that Brown initially includes S (the set of 
involutions generating the Weyl group) as part of the data 
of a BN-pair, he proves later in the chapter that S is 
uniquely determined (as the set of w in W such that 
B union BwB  is a subgroup of G).  So you were right: 
S is described by *properties* of a BN-pair. 

So for us the data will be  (G, B, N)  where B and N are 
subgroups which jointly generate G, and  T = B intersect N 
is normal in N.  Define W := N/T.  Before stating the Hecke 
algebra-based axioms, I'll make some general remarks: 

(1)  There is a sup-preserving map between power sets 

           [ ]: 2[W] --> 2[B\G/B]

sending w to  BwB  (where  BwB := Bw'B  for any w' in N which 
maps to w under the projection  N --> W).  I claim that even under 
the minimal hypotheses given above, this is a lax homomorphism 
from the Boolean group algebra to the Boolean Hecke algebra. 

To see this more clearly, let's express the Boolean Hecke algebra 
structure directly in terms of  2[B\G/B],  starting from the formula 

             J_[k] (gB)  =  Sum_(g' in gB)  g'kB 

(the normalizing factor  1/|B|  being absorbed in the Boolean case). 
Composing, we obtain 

     J_[k'] J_[k] (gB)  =  J_[k'] Sum_(g' in gB)  g'kB

                             =  Sum_(g' in gB)  J_[k'] (g'kB)

                             =  Sum_(g' in gB)  Sum_(g'' in g'kB)  g''k'B

                             =  Sum_(g' in gB)  [ Sum_(b in B)  g'kbk'B]

which makes it clear that 

     J_[k'] J_[k]  =  Sum_(b in B)  J_[kbk']. 

Let me abbreviate the right side to  J_[kBk'],  and the last equation 

      [k].[k']  =  [kBk']  :=  Sum_(b in B)  [kbk'] 

where  [k]  denotes the orientation  BkB  for any element k in G, 
and  [k].[k']  denotes multiplication in  2[B\G/B],  here taken 
as the multiplication which is *opposite* to composition of the 
random jump operators  J_[k].  We have 

    [ww'] <= [wBw'] = [w].[w']

so that  [ ]: 2[W] --> 2[B\G/B]  is a (normal) lax homomorphism. 

(If you don't like passing to the opposite of the algebra of random 
jump operators, another option is to consider  [ ]  as a lax 
homomorphism from  2[W^{op}]  and use the inversion isomorphism 
2[W] ~ 2[W^{op}].  But I'll stick to how I'm doing it here.)  

(2)  We will define the subset S of W to be the set of elements w of 
W for which 

               [w]^2 = [w] + 1  

is a minimal polynomial equation satisfied in the Boolean Hecke 
algebra.  (This means [w] + 1  is a nontrivial idempotent which, 
translated into Brown's language, means that BwB union B  is a 
submonoid of  G.  Under the BN-pair axiom that each such w is 
an involution in W,  this is actually a subgroup of G.) 

BN-pair axioms

(1)  The subset S defined above is a set of involutions which 
generates W.

(2)  For s in S and w in W, the following holds in the Boolean Hecke 
                   [s].[w] <= [w] + [sw]. 

I hope these conditions seem motivated and intuitively reasonable. 
Axiom (2) says that when we start with a flag f in the orientation 
class described by [w] (oriented with respect to a flag stabilized 
by B) and apply a random jump which changes a single feature of 
f corresponding to [s], either we stay in the same orientation class, 
or we jump into a Bruhat cell or chamber which is s-adjacent to the 
one described by [w].  

Remark: These axioms are not quite the same as those given by 
Brown/Tits.  In particular, the assumption that the elements of S 
are involutions, and the assumption that their minimal polynomial 
equations in the Hecke algebra are given as above, may be dropped, 
just by adding the assumption that there exists a set S of generators 
of W such that axiom (2) is satisfied, and also that  sBs^{-1}  is not 
contained in B for any S.  The equations  s^2 = 1  in the group 
algebra and  s^2 = s + 1  in the Hecke algebra are derivable from 
these assumptions (and S is uniquely determined as the set of 
elements obeying these equations). 

As usual in axiomatics, the degree to which one applies Occam's 
razor is a matter of personal aesthetics -- at the expense of some 
extra whittling down, I have chosen a formulation which emphasizes 
the interactions between the group and Hecke algebras.  In practice 
it is probably no harder to verify the axioms as given here than it is 
to verify the Brown/Tits axioms, so the difference is probably harmless. 


(a)  One consequence is a complete description of parabolic 
subgroups (i.e. subgroups intermediate between B and G -- 
Brown actually reserves the word 'parabolic' for a subgroup 
conjugate to one between B and G) on the basis of the axioms.  
First we show that unions of double cosets  BwB,  where w 
ranges over words generated by a subset S' of S, are parabolic 
subgroups.  Or, translated into the language of Hecke algebras, 

                  Sum_(w in )  [w] 

is idempotent for each subset S' of S.  (The idempotency says 
that the union of such double cosets [w] is the submonoid 
generated by B and  {s: s in S'},  but since elements s in S are 
involutions, it's actually a subgroup.) 

It suffices that 

              [w].[w']  <=  Sum_(w in )  [w] 

whenever  w, w'  belong to  .  This is shown by induction on 
the length of a word  used to write w.  The case  d = 0 
is obvious.  The inductive step is an easy consequence of the 
lax homomorphism and Axiom (2): 

   [s1(].[w']  <=  [s1].[].[w']    (lax homomorphism) 

                       <=  [s1]. Sum_(w in )  [w]  (inductive hypothesis)

                       <=  Sum_(w in )  [w] + [(s1)w]  (axiom (2))

                        =  Sum_(w in ) [w].   QED

In particular, the union of the double cosets [w] over all w in W 
(which Brown writes as  BWB)  is a subgroup of G,  and since 
B and N jointly generate G, this subgroup must be G. 

I will save for later the proof of 

Proposition 1.  All intermediate subgroups between B and G are 
described by idempotents in the Hecke algebra of this form.  (Hence 
they are in one-one correspondence with subsets S' of S.) 

(b)  Our next task will be to establish the Bruhat decomposition 
of G.  We had just observed that the union of the double cosets 
BwB  over all w in W  is G,  i.e. that  [ ]: 2[W] --> 2[B\G/B]  is 
surjective.  Now we prove this map is injective, i.e. that  [w] = [w'] 
implies  w = w'. 

The proof is by induction on  d = min{d(w), d(w')}  where *d(w)* is 
the minimum length of a word in S which evaluates to w.  We may 
assume  d = d(w').  

If d = 0, then w' = 1, and if [w] = 1, i.e. if  Bw'B = B  for some w' 
which maps to w under  N --> W,  then w' is in B and hence w = 1
by definition of W.

If d > 0, write w' in the form  sw''  where s is in S and  d(w'') = d-1. 
By assumption,  [sw''] = [w]  and hence 

           [w''] = [s^2 w''] <= [s].[sw''] = [s].[w] 
                                                <= [w] + [sw]    by axiom (2)

and since distinct elements of the form [w] are disjoint, either 
[w''] = [w]  or  [w''] = [sw].  By induction,  w'' = w  or  w'' = sw. 
The first case is impossible since  d(w'') < d(w).  Hence  w'' = sw, 
whence  w' = sw'' = w  (using the fact that s is an involution).  This 
completes the proof.  

(c)  Now we prove a result which is used in the proof that the 
Hecke algebra presentation (recalled below) is correct: 

Lemma 1:  If  d(sw) >= d(w),  then  [s].[w] = [sw]. 

Proof: By induction on  d = d(w).  The case d = 0 is clear.  For 
d > 0,  write  w = w't  where t is in S and  d(w') = d-1.  We will show 
                ([s].[w]) /\ [w] = 0

so that  [s].[w] <= [w] + [sw]  (axiom (2))  implies  [s].[w] = [sw]. 


    d(w') + 1  =  d(w)  <=  d(sw)  =  d(sw't)  <=  d(sw') + 1

(where the first inequality is the hypothesis of the lemma), we 
get  d(w') <= d(sw').  Therefore, by inductive hypothesis,  

                 [sw'] = [s].[w']. 

Now we calculate 

     ([s].[w]) /\ [w]  =  ([s].[w't]) /\ [w] 

                         <= ([s].[w'].[t]) /\ [w]   (lax homomorphism)

                          =  ([sw'].[t]) /\ [w]     (equation above) 

                         <= ([sw'] + [sw't]) /\ [w]   (variant of axiom (2))

                          = ([sw'] + [sw]) /\ [w]

                          = ([sw'] /\ [w]) + ([sw] /\ [w])

The proof is complete if both summands in the last line are 0. 
Because [ ] is injective (from (b) above), it suffices that  sw != w 
(clear) and that  sw' != w  -- this follows from w' != sw,  which 
we know from  d(w') < d(w) <= d(sw).  QED

(d)  Now we prove our principal theorem on Hecke algebra 

Theorem:  If  is a minimal word in S which evaluates to w, 
then  [s1].[s2].(...).[sd] = [w]  in the Boolean Hecke algebra. 

Proof: By induction on d.  The case d = 1 is trivial.  Now 
is a minimal word only if  is a minimal word, so 

        [s2].(...).[sd] = []

by inductive hypothesis.  Since  d( = d-1 < d(  by 
definition of minimality, the hypothesis of Lemma 1 is satisfied, 
so that 

        [s1].[]  =  [] = [w] 

and the inductive step goes through.  QED

Corollary: Under the hypothesis of the preceding theorem, 
[s1].(...).[sd] = [w]  in the Hecke algebra taken over any 
Q_(+)-algebra as base rig. 

Proof:  It suffices to consider the case where the base rig is Q_(+). 
Writing out  [s1].(...).[sd]  as a linear combination of basis elements 
[w] in  B\G/B, 

            [s1].(...).[sd]  =  Sum_(v in W)  a_v [v]      (a_v  in Q_(+)), 

we see by applying the change-of-base-rig  Q_(+) --> 2  that the 
only nonzero  a_v  occurs when  v = w  (using the theorem).  This 
coefficient  a_v = 1  by conservation of probability.  QED

(e)  In serious applications of the theorem and corollary of (d), we 
need to take advantage of the fact that (W, S) is a Coxeter system.
To this end we first prove a companion to Lemma 1: 

Lemma 2:  If  d(w) >= d(sw),  then  [s].[w] = [w] + [sw]. 

Proof:  Since  d(ssw) >= d(sw),  Lemma 1 gives 

                  [w] = [ssw] = [s].[sw]

so that 
                [s].[w]  =  [s]^2 .[sw]

                         =  ([s] + 1).[sw]    (defining property of S)

                         =  [s].[sw]  +  [sw]

                         =  [w]  +  [sw].      QED

To prove that  (W, S)  is a Coxeter system, we verify a somewhat 
technical necessary and sufficient condition called the "folding 
condition" (which I don't understand yet, but which intuitively has 
to do with "folding" a Coxeter complex along a wall, i.e. onto the 
half-space on the side of the wall which contains the "preferred" 
chamber stabilized by B): 

Folding Condition:  Given w in W and s, t in S such that 

              d(sw) = d(w) + 1 = d(wt), 

either  d(swt) = d(w) + 2  or  swt = w. 

Verification:  Suppose  d(swt) < d(w) + 2.  Then by Lemmas 1 and 2, 

                  [s].[w].[t]  =  [s].[wt]  =  [wt] + [swt]

                  [s].[w].[t]  =  [sw].[t]  =  [sw] + [swt]. 

Using the injectivity of  [ ]: 2[W] --> 2[B\G/B],  the summands in 
each of these equations are disjoint, and it follows that  [wt] = [sw], 
whence (again by injectivity)  wt = sw.  Therefore  swt = w.  QED

Having proven that  (W, S)  is a Coxeter system, we may give a 
presentation of the Hecke algebra (over any Q_(+)-algebra), at 
least in the case where G is finite.  Here the minimal polynomial 
of the orientation [s] for s in S is 

             [s]^2 = (1/q)((q-1)[s] + [1])

where q is the cardinality of the Bruhat cell which is s-adjacent to 
the 1-point identity cell B in the "flag manifold" G/B.  Aside from 
these, the remaining equations of the presentation are of the form 

          [s].[t].[s]...  =  [t].[s].[t]...

where  sts... and tst... are alternating words of length  m(s, t)  in 
letters s, t in S,  and where  m(s, t)  is the order of st in W.  (To 
apply the theorem above, we need to know these alternating words 
are minimal; this follows from Tits's solution of the word problem for 
the presentation.) 

I am not quite sure what modifications might be needed when G 
is non-finite, particularly with analogues of the quadratic equations 
for [s]. 

(f)  Finally we return to Proposition 1, on the characterization of 
parabolic subgroups. 

Lemma 3:  If  is a minimal word which evaluates to w, then 
the subgroup of G generated by w and B contains the elements 
s1, ..., sd.  Therefore (Proposition 1) every subgroup intermediate 
between B and G is of the form  

        Sum_(w' in )  [w']   =   Sum_(w' in )  Bw'B 

for some subset S' of S. 

Proof:  By induction on d.  The case d = 0 is trivial.  For  d > 0,  is a minimal word for  (s1)w,  so by inductive hypothesis 
{s2, ..., sd}  is contained in  <(s1)w, B>,  and also we have  
d(s1 w) < d(w).  Hence Lemma 2 applies: 

            [s1].[w] = [w] + [s1 w]

           [s1].[w] /\ [w]  !=  0. 

In other words,  B(s1)BwB  and  BwB  have a common element. 
It follows that  s1  belongs to    and also (as noted above) 

  {s2, ..., sd}  is contained in  <(s1)w, B>  

                    is contained in   =  

which completes the inductive step.  

For the second statement of the lemma, let P be a subgroup 
between B and G.  P is a union of double cosets, i.e. there is a 
uniquely determined subset (in fact a subgroup) T of W such that 

               P  =  Sum_(w in T)  [w]

and if we let S' be the set of s in S which occur in a minimal word 
of any w in T, then the first statement shows S' is contained in P, 
and therefore  T = .     QED

I think I'll stop here for now.  I hope this hasn't made your eyes 
glaze over too much! 


© 2005 Todd H. Trimble