The center of the affine BMW algebra Wk

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last updates: 24 April 2011

The center of the affine BMW algebra Wk

The affine BMW algebra Wk is an algebra over the commutative ring C and the polynomial ring C[ Y1±1 ,, Yk±1 ] is a subalgebra of Wk. The symmetric group Sk acts on C[ Y1±1 ,, Yk±1 ] by permuting the variables and the ring of symmetric functions is C[ Y1±1 ,, Yk±1 ] Sk = { f C[y1,, yk] | wf=f, for wSk }. A classical fact (see, for example [GV, Proposition 2.1]) is that the center of the affine Hecke algebra Hk is Z(k) = C[ Y1±1 ,, Yk±1 ] Sk . The following theorem gives an analogous characterization of the center of the degenerate affine BMW algebra.

The center of the degenerate affine BMW algebra Wk is Rk = {f C[ Y1±1 ,, Yk±1 ] Sk | f(Y1, Y1-1, Y3, ,Yk) = f(1,1,Y3, ,Yk)}.

Step 1: fWk commutes with all yi f C[ Y1±1 ,, Yk±1 ] :
Assume fWk and write f= cd n1,, nk Td n1,, nk in terms of the basis in Theorem 3.1?????????. Let dDk with the maximal number of crossings such that cd n1,, nk 0 and suppose there is an edge (i,j) of d such that ji. Then, by (4.22) and (4.23), the coefficient of Yi Td n1,, nk in Yif is cd n1,, nk and the coefficient of Yi Td n1,, nk in fYi is 0. If Yif=fYi, it follows that there is no such edge, and so d=1 (and therefore Td=1). Thus f C[ Y1±1 ,, Yk±1 ] . Conversely, if f C[ Y1±1 ,, Yk±1 ] then Yif=fYi.

Step 2. f C[ Y1±1 ,, Yk±1 ] commutes with all Ti fRk :
Assume f C[ Y1±1 ,, Yk±1 ] and write

f= a,b Y1a Y2b fa,b,    where fa,b C[ Y3±1 ,, Yk±1 ] .
Then f(1,1, Y3,Yk) = a,b fa,b and
f( Y1, Y1-1, Y3,, Yk) = a,b Y1a-b fa,b = Y1 ( b f+b,b ) . (4.25)
By direct computation using (3.31) and (3.33),
T1 Y1a Y2b = Y1a Y1a T1 Y2b-a = s1( Y1a Y2b ) T1 + (q-q-1) Y1a Y2b - s1( Y1a Y2b ) 1 - Y1 Y2-1 + b-a ,
= { -(q-q-1) r=1 Y1-r E1 Y1-r, if>0, (q-q-1) r=1 - Y1+r-1 E1 Y1r-1, if<0, 0, if=0.
It follows that
T1 f = (s1f) T1 + (q-q-1) f-s1f 1- Y1 Y2-1 + 0 ( b f +b,b ) . (4.26)
Thus if f(Y1, Y1-1, Y3, ,Yk) = f(1,1,Y3, ,Yk) then, by (4.25),
b f +b,b =0, for 0 . (4.27)
Hence, if f C[ Y1±1 ,, Yk±1 ] Sk and f(Y1, Y1-1, Y3, ,Yk) = f(1,1,Y3, ,Yk) then s1f=f and (4.27) holds so that, by (4.26), T1f= fT1 . Similarly, f commutes with all Ti.
Conversely, if f C[ Y1±1 ,, Yk±1 ] and Tif= fTi then
sif=f     and     b f +b,b =0, for 0 ,
so that f C[ Y1±1 ,, Yk±1 ] Sk and f(Y1, Y1-1, Y3, ,Yk) = f(1,1,Y3, ,Yk) . □

The symmetric group Sk acts on k by permuting the factors. The ring

C[ Y1±1 ,, Yk±1 ] Sk     has basis     {mλ | λk with λ1 λ2 λk },
mλ = γ Skλ Y1γ1 Ykγk .
The elementary symmetric functions are
er = m(1r, 0k-r) and e-r = m(0k-r, (-1)r ) ,for r=0,1,,k,
and the power sum symmetric functions are
pr = m(r, 0k-1) and p-r = m(0k-1, -r) ,for r0 .
The Newton identities (see [Mac, Ch. I (2.11′)]) say
e = r=1 (-1)r-1 pr e-r     and     e- = r=1 (-1)r-1 p-r e-(-r) ,
where the second equation is obtained from the first by replacing Yi with Yi-1. For and λ=(λ1, ,λk) k,
ek mλ = mλ +(k) ,     where  λ +(k) =(λ1+, , λk+) .
In particular,
e-r = ek-1 ek-r, for r=0,1,k . (4.29)
pi+ = pi+p-i , pi- = pi-p-i, for i>0 .
The consequence of (4.29) and (4.28) is that
[ Y1±1 ,, Yk±1 ] Sk = [ ek±1, e1,, ek-1] = [ ek±1] [e1, e2, ek2 , ek e- k-12 , eke-2, eke-1 ] = [ ek±1] [e1, e2, ek2 , e- k-12 , e-2, e-1 ] = [ ek±1] [p1, p2, pk2 , p- k-12 , p-2, p-1 ] = [ ek±1] [ p1+, p2+, pk2 + , p k-12 - , p2-, p1- ].
For νk with ν1 ν>0 define
pν+ = pν1+ pν+ and pν- = pν1- pν- .
C[ Y1±1 ,, Yk±1 ] Sk has basis { ek pλ+ pμ- | , (λ) k2 , (μ) k-12 } . (4.29)
In analogy with (4.12) we expect that if Rk is as in Theorem 4.6 then
Rk = C[ ek±1] [ p1-, p2-, ] . (4.30)

The left ideal of W2 generated by E1 is C[ Y1±1 ] E1 . This is an infinite dimensional (generically irreducible) W2-module on which Z(W2) acts by constants. Thus, as noted by IS THIS IN RUI SI?????, it follows that W2 is not finitely generated as a Z(W2)-module.

Notes and references

This page is the result of joint work with Zajj Daugherty and Rahbar Virk [DRV]. The characterisation of the center of the affine BMW algebra given here is analogous to that for the degenerate affine BMW algebra as found in (???). PUT A REMARK ABOUT the solution of [St. Exercise 7.7] as found on [St., p. 497] in THIS CONTEXT???


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