Examples of Macdonald polynomials type A1

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 23 December 2012

Type A1

The Weyl group W0= s1 s12=1 has order two and acts on the lattices 𝔥=ωand 𝔥*=ω bys1ω=- ωands1 ω=-ω, (4.1) and φ= α =2ω, φ= α=2ω, and ω,α=1. (4.2) gs0s1s0 gs0s1 gs0 g gs1 gs1s0 gs1s0s1 X-3ω X-ωs1 X-ω Xωs1 Xω X3ωs1 X3ω s1s0s1 s1s0 s1 1 s0 s0s1 s0s1s0 X-αs1 X-α s1 1 Xαs1 Xα X2αs1 𝔥α+2d 𝔥α+d 𝔥α 𝔥-α+d 𝔥-α+2d 𝔥-α+3d The double affine braid group is generated by T0, T1, g, Xω, and q 12, with relations T0= gT1g-1, g2=1, q=Xδ, gXω= q12 X-ωg, T1 Xω T1 = X-ω, and T0 X-ω T0 = q-1 Xω. (4.3) In the double affine braid group g= Yω T1-1 , T0 = Yφ T1-1 , g = Xω T1 , (T0) 1 = Xφ T1. (4.4) At this point, the following Proposition, which is the Type A1 case of Theorem 2.1, is easily proved by direct computation.

(Duality). Let Yd= q-1. The double affine braid group is generated by T0, T1, g, Yω and q12 with relations Yd =q-1 , (g)2 =1, T0 =g T1 (g) -1, gYω = q-12 Y-ω g , T1-1 Yω T1-1 ,and (T0) -1 Y-ω (T0) -1 =qYω.

Proof.

We prove that the presentation in (5.7) is equivalent to the presentation in (5.3). The proof that the presentation in (5.6) is equivalent to the presentation in (5.3) is similar.

(5.3)(5.7): Use (5.4) to define Yω in terms of g and T1. The first and second relations in (5.7) are the third and fourth relations (5.3). The proof of the third, fourth and fifth relations in (5.7) are

gYω= ggT1= q-12 T1-1gg= q-12 Y-ωg, T1-1Yω T1-1=T1-1 gT1T1-1= T1-1g= Y-ω,

and

(T0)-1 Y-ω (T0)-1 =gT1-1 gY-ω gT1-1g =q12g T1-1 YωT1-1 g=q12 gY-ω g=qYω,

respectively.

(5.7)(5.3): Define g=Yω T1-1 and T0=Y2ω T1-1. The third and fourth relations of (5.3) are exactly the first and second relations of (5.7). The proof of the first, second and fifth relations in (5.3) are

g2=Yω T1-1Yω T1-1= Yω Y-ω=1, gT1g-1= YωT1-1 T1T1 Y-ω= YωT1 Y-ω= YωT1 Y-ωT1 T1-1= Y2ω,

and

T1gg=T1g YωT1-1 =q-12T1 Y-ωg T1-1= q-12gg T1-1,

respectively.

The double affine Hecke algebra H is with the additional relations Ti2= ( t1/2- t-1/2 ) Ti+1 ,for i=0,1, and t0= t1= t= qc. (4.5)

Using (4.5), the relations in Proposition 4.1 give gYω= q-1/2 Y-ω g , T1Yω= Y-ωT1+ ( t1/2- t-1/2 ) Yω- Y-ω 1-Y-α ,and T0Yω= q-1Y-ω T0+ ( t1/2- t-1/2 ) ( Yω-q-1 Y-ω 1-qYα ) . With Yα0 =qYα and Yα1 =Yα, then τg=g,and τi=Ti- ( t1/2- t-1/2 ) ( 1 1-Y-αi ) ,fori=0,1.

To illustrate Theorem 2.2, note that X-2ω = s1s0 is a reduced word and

τ1τ0 = ( T1+ t-1/2(1-t) 1-Y-α1 ) τ0 = T1T0+ T1 t-1/2(1-t) 1-Y-α0 +(T0)-1 t-1/2(1-t) 1-Y-s0α1 + ( t-1/2(1-t) 1-Y-s0α1 ) ( t-1/2(1-t) Y-α0 1-Y-α0 ) = X-2ω + T1 t-1/2(1-t) 1-Y-α0 +X2ω T1 t-1/2(1-t) 1-Y-s0α1 + ( t-1/2(1-t) 1-Y-s0α1 ) ( t-1/2(1-t) Y-α0 1-Y-α0 ) The corresponding paths in (1,-2ω) =B(-2ω) are X-2ω T1 t-1/2(1-t) 1-Y-α0 X2ω T1 t-1/2(1-t) 1-Y-s0α1 t-1/2(1-t) 1-Y-s0α1 t-1/2(1-t) Y-α0 1-Y-α0

The polynomial representations is defined by g1=1,T0 1=t121, andT11= t121. In this case ρc=12cα andW0= { X-w 0 } { Xωs1 >0 } , (4.6) is the set of minimal length coset representatives of W/W0.

Applying the expansion of τ1τ0 to 1 and using

Y-α01=q Yα1=qqc 1=qt1,and Y-s0α11= Yα+2d1= q2Yα1= q2t,

gives

E-2ω = τ1τ01 = X-2ω+t1/2 t-1/2(1-t) 1-qt +X2ωt1/2 t-1/2(1-t) 1-q2t + ( t-1/2(1-t) 1-q2t ) ( t-1/2(1-t)qt 1-qt ) = X-2ω+ 1-t1-qt +X2ω 1-t1-q2t+ (1-t1-q2t) ((1-t)q1-qt) . Since 10=T1+t-1/2 the symmetric Macdonald polynomial P2ω=10 E2ω=10 τ01 is

P2ω = 10E2ω= (T1+t-1/2) τ01 = ( T1T0+ T1 t-1/2(1-t) 1-Y-α0 +t-1/2 (T0)-1+ t-1/2 t-1/2(1-t) Y-α0 1-Y-α0 ) 1 = ( X-2ω+ t1/2 t-1/2(1-t) 1-qt +t-1/2 X2ωT1+ t-1/2 t-1/2(1-t) qt 1-qt ) 1 = ( X-2ω+ t1/2 1-t 1-qt +X2ω+ t-1/2 (1-t) q 1-qt ) 1 = ( X2ω+ X-2ω+ t1/2+ (1+q) 1-t 1-qt ) 1. The corresponding paths in 𝒫(2ω) are X-2ω 1-t1-tq X2ω q (1-t)1-tq

Notes and References

This page is section 4.1 from the paper of A. Ram and M. Yip entitled A combinatorial formula for Macdonald polynomials.

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