The center of degenerate affine BMW algebra 𝒲k

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

Last updates: 23 April 2011

The center of the degenerate affine BMW algebra 𝒲k

The degenerate affine BMW algebra 𝒲k is an algebra over the commutative ring C and the polynomial ring C[y1,, yk] is a subalgebra of 𝒲k. The symmetric group Sk acts on C[y1,, yk] by permuting the variables and the ring of symmetric functions is C[y1,, yk] Sk = { f C[y1,, yk] | w p=p, for wSk }. A classical fact (see, for example [Kl, Theorem 3.3.1]) is that the center of the degenerate affine Hecke algebra k is Z(k) = C[y1,, yk] 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 k = {f C[y1,, yk] Sk | f(y1, -y1,y3, ,yk) = f(0,0,y3, ,yk)}.

Proof. Step 1: f commutes with all yi f C[y1,, yk] :
Assume f𝒲k and write f= cd n1,, nk d 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 (3.5) and (3.6) (ELSEWHERE) the coefficient of yi d n1,, nk in yif is cd n1,, nk and the coefficient of yi d n1,, nk in fyi is 0. If yif=fyi, it follows that there is no such edge, and so d=1. Thus f C[y1,, yk]. Conversely, if f C[y1,, yk] then yif=fyi.

Step 2. f C[y1,, yk] commutes with all tsi fk :
Assume f C[y1,, yk] and write

f= a,b 0 y1a y2b fa,b,    where fa,b C[y3,, yk].
Then f(0,0, y3,yk) = a,b 0 fa,b and
f( y1,-y1, y3,, yk) = a,b 0 (-1)b y1a+b fa,b = 0 y1 ( b=0 (-1)b f-b,b ) . (4.9)
By direct computation using (3.11) and (3.12),
ts1 y1a y2b = s1( y1a y2b ) ts1 - y1a y2b - s1( y1a y2b ) y1 - y2 +(-1)a r=1 a+b (-1)r y1a+b-r e1 y1r-1 ,
and it follows that
ts1 f = (s1f) ts1 - f-s1f y1 - y2 + >0 ( ( r=1 (-1)r y1-r e1 y1r-1 ) ( b=0 (-1)-b f -b,b ) ) . (4.10)
Hence, if f C[y1,, yk] Sk and f(y1, -y1,y3, ,yk) = f(0,0,y3, ,yk) then s1f=f and, by (4.9), (4.11) holds so that, by (4.10), ts1 f=f ts1 . Similarly, f commutes with all tsi .
Conversely, if f C[y1,, yk] and tsi f=f tsi then
sif=f     and     b=0 (-1)-b f -b,b =0 ,   for  0,
so that f C[y1,, yk] Sk and f(y1, -y1,y3, ,yk) = f(0,0,y3, ,yk) . □

The power sum symmetric functions pi, i>0, are given by

pi = y1i + y2i ++ yki .
The Hall-Littlewood polynomials (see [Mac, Ch. III (2.1)]) are given by
Pλ(y;t) = Pλ( y1,, yk;t) = 1vλ(t) wSk w( y1λ1 ykλk 1i<jk xi-txj xi-xj ) ,
where vλ(t) is a normalizing constant (a polynomial in t) so that the coefficient of y1λ1 ykλk in Pλ(y;t) is equal to 1. The Schur Q-functions (see [Mac, Ch. III (8.7)]) are
Qλ = { 0, ifλ is not strict, 2(λ) Pλ(y;-1) , ifλ is strict,
where (λ) is the number of parts of λ and the partition λ is strict if all its parts are distinct. Let k be as in Theorem 4.2. Then (see [Naz, Cor. 4.10], [Pr, Theorem 2.11(Q)] and [Mac, Ch. III §8])
k = C[p1, p3, p5,] = C-span{ Qλ | λis strict } . (4.12)
More generally, let r >0 and let ζ be a primitive rth root of unity. Define
r,k = {f [ζ] [y1,, yk] Sk | f( y1, ζy1, ζr-1 y1, yr+1 ,, yk) = f(0,0,0, yr+1 ,, yk) } .
r,k [ζ] (ζ) = (ζ) [pi | i0modr] , (4.13)
and r,k has [ζ]-bases
{ Pλ(y;ζ) | mi(λ) <r and λ1k} and { qλ(y;ζ) | mi(λ) <r and λ1k} , (4.14)
where mi(λ) is the number of parts of size i in λ. The ring r,k is studied in [Mo], [LLT], [Mac, Ch. II Ex. 5.7 and Ex. 7.7], [To], [FJ+], and others. The proofs of (4.13) and (4.14) follow from [Mac, Ch. III Ex. 7.7.], [To, Lemma 2.2 and following remarks] and the arguments in the proofs of [FJ+, Lemma 3.2 and Proposition 3.5].

Notes and references

This page is the result of joint work with Zajj Daugherty and Rahbar Virk [DRV]. Nazarov give a characterization of Z(𝒲k) in [Naz, Cor 4.10]. the solution of [St. Exercise 7.7] as found on [St., p. 497],


[AMR] S. Ariki, A. Mathas, and H. Rui, Cyclotomic Nazarov Wenzl algebras, Nagoya Math. J. 182, (2006), 47-134. arXiv:math/0506467, MR2235339

[DRV] Z. Daugherty, A. Ram, and R. Virk, Affine and graded BMW algebras, in preparation.

[FJ+] B. Feigin, M. Jimbo, T. Miwa, E. Mukhin, and Y. Takeyama, Symmetric polynomials vaishing on the diagonals shifted by roots of unity, Int. Math. Res. Notices 2003 no. 18, 1015-1034, arXiv:math/0209126. MR??????

[LLT] A. Lascoux, B. Leclerc, J.-Y. Thibon, Green polynomials and Hall-Littlewood functions at roots of unity, Europ. J. Combinatorics 15 (1994), 173-180. MR??????

[Mac] I.G. Macdonald, Symmetric functions and Hall polynomials, Second edition, Oxford University Press, 1995. ISBN: 0-19-853489-2 MR1354144

[Mo] A.O. Morris, On an algebra of symmetric functions, Quart. J. Math. 16 (1965), 53-64. MR??????

[Pr] P. Pragacz, Algegro-geometric applications of Schur S and Q polynomials in Topics in invraiant theory (Paris, 1989/1990), Lecture Notes in Math. 1478, Springer, Berlin (1991), 130-191. MR1180989

[St] R.P. Stanley, Enumerative Combinatorics Vol. 2 Cambridge Univ. Press 1999. ISBN: 0-521-56069-1; 0-521-78987-7 MR1676282

[To] B. Totaro, Towards a Schubert calculus for complex reflection groups, Math. Proc. Camb. Phil. Soc. 134 (2003), 83-93. MR??????

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