Kernel function and quantum algebras

KERNEL FUNCTION AND QUANTUM ALGEBRAS

arXiv:1002.2485v1 [math.QA] 12 Feb 2010

B. FEIGIN, A. HOSHINO, J. SHIBAHARA, J. SHIRAISHI AND S. YANAGIDA Abstract. We introduce an analogue Kn (x, z; q, t) of the Cauchy-type kernel function for the Macdonald polynomials, being constructed in the tensor product of the ring ΛF of symmetric functions and the commutative algebra A over the degenerate CP1 . We show that a certain restriction of Kn (x, z; q, t) with respect to the variable z is neatly described by the tableau sum formula of Macdonald polynomials. Next, we demonstrate that the level m representation of the Ding-Iohara quantum algebra U(q, t) naturally produces the currents of the deformed Wq,p (sln ). Then we remark that the Kn (x, z; q, t) emerges in the highest-to-highest correlation function of the deformed Wq,p (sln ) algebra.

1. Kernel function 1.1. The algebra A. We briefly recall the definition and the basic facts about the commutative algebra A introduced in [FHHSY]. Let q1 , q2 be two independent indeterminates and set q3 := 1/q1 q2 . We also use the symbols F := Q(q1 , q2 ), N := {0, 1, 2, . . .} and N+ := {1, 2, . . .}. For n, k ∈ N+ , we define two operators ∂ (0,k) , ∂ (∞,k) acting on the space of symmetric rational functions in n variables x1 , . . . , xn by ∂ (0,k)

: f

7→

∂ (∞,k) : f

7→

n! lim f (x1 , . . . , xn−k , ξxn−k+1 , ξxn−k+2 , . . . , ξxn ) (n − k)! ξ→0 n! lim f (x1 , . . . , xn−k , ξxn−k+1 , ξxn−k+2 , . . . , ξxn ) (n − k)! ξ→∞

whenever the limit exists. We also set ∂ (0,k) c = 0, ∂ (∞,k) c = 0 for c ∈ F. Finally we define ∂ (0,0) and ∂ (∞,0) to be the identity operator. Definition 1.1. For n ∈ N, the vector space An = An (q1 , q2 , q3 ) is defined by the following conditions (i), (ii), (iii) and (iv). (i) A0 := F. For n ∈ N+ , f (x1 , . . . , xn ) ∈ An is a rational function with coefficients in F, and symmetric with respect to the xi ’s. (ii) For n ∈ N, 0 ≤ k ≤ n and f ∈ An , the limits ∂ (∞,k) f and ∂ (0,k) f both exist and coincide: ∂ (∞,k) f = ∂ (0,k) f (degenerate CP1 condition). (iii) The poles of f ∈ An are located only on the diagonal {(x1 , . . . , xn ) | ∃(i, j), i 6= j, xi = xj }, and the orders of the poles are at most two. (iv) For n ≥ 3, f ∈ An satisfies the wheel conditions f (x1 , q1 x1 , q1 q2 x1 , x4 , . . .) = 0,

f (x1 , q2 x1 , q1 q2 x1 , x4 , . . .) = 0. L Then we set the graded vector space A = A(q1 , q2 , q3 ) := n≥0 An .

Definition 1.2. For an m-variable symmetric rational function f and an n-variable symmetric rational function g, we define an (m + n)-variable symmetric rational function f ∗ g by Y (f ∗ g)(x1 , . . . , xm+n ) := Sym f (x1 , . . . , xm )g(xm+1 , . . . , xm+n ) (1.1) ω(xα , xβ ) . 1≤α≤m m+1≤β≤m+n

Date: Feburary 12, 2010. 1

2

B. FEIGIN, A. HOSHINO, J. SHIBAHARA, J. SHIRAISHI AND S. YANAGIDA

Here ω(x, y) is the rational function (x − q1 y)(x − q2 y)(x − q3 y) , (x − y)3 P and the symbol Sym means Sym(f (x1 , . . . , xn )) := (1/n!) σ∈Sn f (xσ(1) , . . . , xσ(n) ). ω(x, y) = ω(x, y; q1 , q2 , q3 ) :=

(1.2)

Fact 1.3 ([FHHSY, Theorem 1.5]). A is closed with respect to ∗, and the pair P Q (A, ∗) is a unital associative commutative algebra. The Poincar´e series is n≥0 (dimF An )z n = m≥1 (1 − z m )−1 .

1.2. The ring ΛF of symmetric functions. As for the notations and definitions concerning the partitions, we basically follow the notation in [M]. A partition of n ∈ N is a sequence λ = (λ1 , λ2 , . . .) of non-negative integers satisfying λ1 ≥ λ2 ≥ · · · . We define |λ| := λ1 + λ2 + · · · , ℓ(λ) := #{i | λi 6= 0}, and write λ ⊢ n if |λ| = n. We denote the conjugate (transpose) of a partition λ by λ′ . We def

work with the dominance partial ordering defined as : λ ≥ µ ⇐⇒ |λ| = |µ|, λ1 + · · · + λi ≥ µ1 + · · · + µi for all i ≥ 1. We recall some basic facts about the ring of symmetric functions. As was in [FHHSY], we set q1 = −1 q , q2 = t (hence q3 = qt−1 ) and F = Q(q1 , q2 ) = Q(q, t). Let ΛF be the ring of symmetric functions over the base field F, constructed in the category of graded ring with the projection operators ρm,n : f (x1 , . . . , P xm ) 7→ f (x1 , . . . , xn , 0 . . . , 0). Let pn (x) := i xni be the power sum P function. For a partition λ = (λ1 , λ2 , . . .), the monomial of λ. symmetric function is defined by mλ (x):= α xα , where α runs over all the distinct permutations Q The elementary symmetric function e (x) is defined by the generating function E(y):= (1+x y) = n i i Q P Q P n . Set G(y):= n , where (x; q) := e (x)y {(tx y; q) /(x y; q) } = g (x; q, t)y (1− n i ∞ i ∞ n ∞ n≥0 i n≥0 i≥0 q i x). For a partition λ = (λ1 , λ2 , . . .) set pλ := pλ1 pλ2 · · · . Similarly we write eλ := eλ1 eλ2 · · · and gλ := gλ1 gλ2 · · · . It is known that {pλ }, {mλ }, {eλ } andQ {gλ } form bases Q of ΛF . Recall Macdonald’s scalar product hpλ , pµ iq,t := δλ,µ i≥1 imi mi ! j≥1 (1 − q λj )/(1 − tλj ), where mi denotes the number of parts i in the partition λ. For any dual bases {uλ } and {vλ }, we have X Y (txi yj ; q)∞ = uλ (x)vλ (y). (1.3) Π(x, y; q, t) := (xi yj ; q)∞ i,j

λ

It is known that {mλ } and {gλ } form dual bases, namely we have hmλ , gµ iq,t = δλ,µ . Macdonald polynomials Pλ (x; q, t) are uniquely characterized by (a) the triangular expansion Pλ = P mλ + µ0 n>0 X 1 − tn (1 − p−n )pn/4 an z −n , ϕ+ (z) := exp − n n>0 X 1 − t−n − −n n/4 n ϕ (z) := exp (1 − p )p a−n z . n n>0

e× , we have a level one representation ρc (·) of U (q, t) on F by setting Then for a fixed c ∈ F ρc (γ ±1/2 ) = p∓1/4 ,

ρc (ψ ± (z)) = ϕ± (z),

ρc (x+ (z)) = c η(z),

ρc (x− (z)) = c−1 ξ(z).

Remark 2.4. We can rephrase this fact as follows. Let us define bn ’s by the expansion of ψ ± : ! ! X X (2.2) b−n γ n/2 z n . ψ + (z) = ψ0+ exp + bn γ n/2 z −n , ψ − (z) = ψ0− exp − n>0

n>0

Then we have 1 (1 − q −m )(1 − tm )(1 − pm )(γ m − γ −m )γ −|m| δm+n,0 , m and the coproduct for bn reads [bm , bn ] =

∆(bn ) = bn ⊗ γ −|n| + 1 ⊗ bn .

(2.3)

(2.4)

Then the representation ρc is given by γ ±1/2 7→ p∓1/4 and bn 7→ −

1 − tn |n|/2 (p − p−|n|/2 )an , |n|

ψ0± 7→ 1,

x+ (z) 7→ c η(z),

x− (z) 7→ c−1 ξ(z).

Definition 2.5. Consider the m-fold tensor representation ρy1 ⊗ · · · ⊗ ρym on F ⊗m for m ∈ Z≥2 . Define ∆(m) inductively by ∆(2) := ∆,

∆(m) := (id ⊗ · · · ⊗ id ⊗ ∆) ◦ ∆(m−1) .

Since we have ρy1 ⊗ · · · ⊗ ρym ∆(m) (γ) = γ(1) · · · γ(m) = p−m/2 , the level is m. We also define ρy(m) := ρy1 ⊗ · · · ⊗ ρym ◦ ∆(m) : U (q, t) → F ⊗m .

(2.5)


9

Lemma 2.6. We have ρy(m) (x+ (z))

=

m X i=1

e i (z), where the Λ

e ∗ (z) Λ i

e i (z), yi Λ

ρy(m) (x− (z))

=

m X i=1

are defined to be

e ∗ (z), yi−1 Λ i

e i (z) := ϕ− (p−1/4 z) ⊗ ϕ− (p−3/4 z) ⊗ · · · ⊗ ϕ− (p−(2i−3)/4 z) ⊗ η(p−(i−1)/2 z) ⊗ 1 ⊗ · · · ⊗ 1, Λ

e ∗i (z) := 1 ⊗ · · · ⊗ 1 ⊗ ξ(p−(m−i)/2 z) ⊗ ϕ+ (p−(2m−2i−1)/4 z) ⊗ · · · ⊗ ϕ+ (p−1/4 z), Λ

(2.6) (2.7)

where η(p−(i−1)/2 z) and ξ(p−(m−i)/2 z) sit in the i-th tensor component. Proof. By the definition (2.5), Fact 2.3 and Remark 2.4.

2.3. New currents t(z) and t∗ (z). Definition 2.7. We define t(z) := α(z)x+ (z)β(z),

t∗ (z) := α(p−1 z)−1 x− (p−1 γ −1 z)β(γ −2 p−1 z)−1 .

Here we used auxiliary vertex operators ∞ X 1 n , b z α(z) := exp − −n γ n − γ −n

1 −n . b z n γ n − γ −n n=1 n=1 P −(2i+1)n . Here the part 1/(γ n − γ −n ) is considered to be the formal power sum ∞ i=0 γ β(z) := exp

∞ X

(2.8)

(2.9)

Remark 2.8. The definition of t∗ (z) can be read as

t∗ (γpz) = α(γz)−1 x− (z)β(γ −1 z)−1 . This form is convenient in the actual calculations. Proposition 2.9. (1) The elements t(z) and t∗ (z) commutes with α(w), β(w) and ψ ± (w): [t(z), α(w)] = [t(z), β(w)] = [t∗ (z), α(w)] = [t∗ (z), β(w)] = 0, [t(z), ψ ± (w)] = [t∗ (z), ψ ± (w)] = 0. (2) Set ∞ X 1 (1 − q n )(1 − t−n )(1 − p−n γ −2n ) n (2.10) z , n 1 − γ −2n n=1 P −2n −2in . Then we have where the part 1/(1 − γ ) is considered to be the formal power sum ∞ i=0 γ

A(z) := exp

(1 − q)(1 − t−1 ) [δ(p−1 wz )t(2) (z) − δ(p wz )t(2) (w)], 1−p (1 − q −1 )(1 − t) A( wz )t∗ (z)t∗ (w) − A( wz )t∗ (w)t∗ (z) = [δ(p wz )t∗(2) (z) − δ(p−1 wz )t∗(2) (w)], 1 − p−1 P P where δ(z) := n∈N z n + z −1 n∈N z −n is the formal delta function, and A( wz )t(z)t(w) − A( wz )t(w)t(z) =

(2.11) (2.12)

t(2) (z) := α(pz)α(z)x+ (pz)x+ (z)β(pz)β(z),

t∗(2) (z) := α(γpz)−1 α(γz)−1 x− (pz)x− (z)β(γ −1 pz)−1 β(γ −1 z)−1 . (3) As in (2), set B(z) := exp

∞ X 1 (1 − q n )(1 − t−n )(p−2n γ −2n − p−n γ −2n ) n z . n 1 − γ −2n n=1

(2.13)

10


Then B( wz )t(z)t∗ (w) − B(γ 2 p2 wz )t∗ (w)t(z) = Proof. See §3.2.

(1 − q)(1 − t−1 ) −1 w + δ(p z )ψ0 − δ(γ −2 p−1 wz )ψ0− ) . 1−p

(2.14)

In the next subsection we show that the currents t(z), t∗ (z) are connected to the realization of deformed W algebra in the Fock representation of U (q, t). 2.4. Deformed algebra Wq,p (slm ). We basically follow the description of Wq,p (slm ) in [FF, §4]. As for the connection between the singular vectors of the Wq,p (slm ) and the Macdonald polynomials, see [SKAO, AKOS]. Definition 2.10. Set fk,l (z) := exp

∞ X (1 − q n )(1 − t−n )(p(k−1)n − p(l−1)n )

1 − pln

n=1

zn . (m)

Remark 2.11. Our functions A(z) and B(z) give special cases of this function under ρy , that is, ρy(m) (A(z)) = f1,m (z),

ρy(m) (B(z)) = fm−1,m (z).

Definition 2.12. Set T (z) = T1 (z) := ρy(m) (t(z)),

T ∗ (z) = T1∗ (z) := ρy(m) (t∗ (z)).

(2.15)

Let us also define e i (z)ρ(m) (β(z)), Λi (z) := ρy(m) (α(z))Λ y

e ∗ (p(m−2)/2 z)ρ(m) (β(γ −2 p−1 z)−1 ). Λ∗i (z) := ρy(m) (α(p−1 z)−1 )Λ i y (2.16)

Then by Definition 2.7 and Lemma 2.6 we have T1 (z) =

m X

yi Λi (z),

i=1

T1∗ (z)

=

m X

yi−1 Λ∗i (z).

(2.17)

i=1

For i = 2, . . . , m, we further define X Ti (z) := yj1 yj2 · · · yji : Λj1 (z)Λj2 (zp) · · · Λji (zpi−1 ) :,

(2.18)

1≤j1 0

If k > j, then the normal order coefficient is (3.23). The product of (3.25), (3.26), (3.27), (3.28), (3.23) is equal to f1,m (w/z)−1 γ+ (z, w; q, p). Thus we obtain the result. The case i > j is similar, so we omit the detail. 3.3.2. Proof of (2). The desired equation is equivalent to ρy(m) (α(z) · · · α(pm−1 z)) :

m Y

k=1

e k (pk−1 z) : ρ(m) (β(z) · · · β(pm−1 z)) = 1. Λ y

We will show this equation by comparing each tensor component. (m) By (3.21), the k-th tensor component of ρy (α(z) · · · α(pm−2 z)) is equal to X1 −n (2m−k−1)n/2 n (1 − t )p a−n z . exp − n

(3.29)

n>0

(m)

Similarly, the k-th tensor component of ρy (β(z) · · · β(pm−1 z)) is equal to X 1 exp (1 − t−n )p(−k+1)n/2 an z −n . n n>0 The k-th tensor component of :

Qm

e

k=1 Λk (p

k−1 z)

(3.30)

: is

: η(p−(k−1)/2 pk−1 z)ϕ− (p−(2k−1)/4 pk z)ϕ− (p−(2k−1)/4 pk+1 z) · · · ϕ− (p−(2k−1)/4 pm−1 z) : X 1 − tn X 1 − t−n (3.31) pn(2m−k−1)/2 a−n z n exp − p−n(k−1)/2 an z −n = exp n n n>0

n>0

It is easy to see that (3.29),(3.30) and (3.31) cancel. Thus we have the consequnce. 3.3.3. Proof of (3). The desired equation is equivalent to ρy(m) (α(p−1 z) · · · α(pm−2 z)) :

i−1 Y

k=1

e ∗i (p(m−2)/2 z). =Λ

e k (pk−1 z) Λ

m Y

l=i+1

e l (pl−2 z) : ρ(m) (β(z) · · · β(pm−1 z)) Λ y

We will show this equation by comparing each tensor component.

(3.32)


19

(m)

As in (3.29) , the k-th tensor component of ρy (α(p−1 z) · · · α(pm−2 z)) is equal to X1 exp − (1 − tn )p(2m−k−3)n/2 a−n z n . n

(3.33)

n>0

(m)

The k-th tensor component of ρy (β(z) · · · β(pm−1 z)) is given by (3.30). Q Q e l−2 z) : depends on k. If k = i, then e k−1 z) m The k-th tensor component of : i−1 l=i+1 Λl (p k=1 Λk (p by Lemma 2.6 and some simple calculations, the component turns out to be ϕ− (p−(2i−1)/4 pi−1 z)ϕ− (p−(2i−1)/4 pi z) · · · ϕ− (p−(2i−1)/4 pm−2 z) X 1 − t−n = exp − p(i−3)/2 (1 − pn(m−i) )a−n z n . n

(3.34)

n>0

Similarly, if k < i, then by Lemma 2.6 the component is

: η(p−(k−1)/2 pk−1 z)ϕ− (p−(2k−1)/4 pk z)ϕ− (p−(2k−1)/4 pk+1 z) · · · ϕ− (p−(2k−1)/4 pm−2 z) : X 1 − tn X 1 − t−n (3.35) pn(2m−k−3)/2 a−n z n exp − p−n(k−1)/2 an z −n : = exp n n n>0 n>0

If k > i, then by Lemma 2.6 the component is

: η(p−(k−1)/2 pk−2 z)ϕ− (p−(2k−1)/4 pk−1 z)ϕ− (p−(2k−1)/4 pk z) · · · ϕ− (p−(2k−1)/4 pm−2 z) : X 1 − t−n X 1 − tn = exp (3.36) pn(2m−k−3)/2 a−n z n exp − p−n(k−3)/2 an z −n : n n n>0

n>0

Then the i-th tensor component of (3.32) is the product of (3.33),(3.30) and (3.34). After a short e ∗ (p(m−2)/2 z). calculation, one finds that it is ξ(p(i−2)/2 z), which is the i-th component of Λ i If k < i, then the k-th tensor component of (3.32) is the product of (3.33),(3.30) and (3.35). It is e ∗ (p(m−2)/2 z). 1, that is, the k-th component of Λ i Finally, for k > i, the k-th tensor component of (3.32) is the product of (3.33),(3.30) and (3.36). e ∗ (p(m−2)/2 z). It turns out to be ϕ− (p(2j−5)/4 z), which is the k-th component of Λ i

3.3.4. h Q Proof of (4). From i the known identities (2.20) and (2.22), it is not difficult to calculate m−1 −k+l w/z) Λ∗i (z)Λ∗j (w) in terms of Λk ’s. k,l=1 f1.m (p First we consider the case i = j. From the operator product (2.20), we have h m−1 Y

k,l=1

=

i f1,m (p−k+l wz ) Λ∗i (z)Λ∗i (w)

h m−2 Y m−1 Y

k=1 l=k+1

ih m−1 i Y k−1 Y γ+ (p−k+l wz ) γ− (p−k+l wz ) : Λ∗i (z)Λ∗i (w) : k=2 l=1

X 1 1 − p−n(m−2) 1 − pn(m−1) w n (1 − q n )(1 − t−n ) : Λ∗i (z)Λ∗i (w) : . = exp n 1 − p−n 1 − pn z n>0

Here we used the abbreviation γ± ( wz ) := γ± (z, w; q, p). Then we have h m−1 Y

k,l=1

i X1 1 − p−n(m−2) 1 − pn(m−1) w n (1 − q n )(1 − t−n ) f1,m (p−k+l wz ) × exp − n 1 − p−n 1 − pn z n>0

X 1 (1 − q n )(1 − t−n )(1 − p(m−1)n w n = f1,m ( wz ). = exp − mn n 1 − p z n>0

20


Thus the desired equation f1,m ( wz )Λ∗i (z)Λ∗i (w) =: Λ∗i (z)Λ∗i (w) : is proved. Next, note that the calculation of the case i 6= j reduces to that of k = i. If i < j, then f1,m ( wz )Λ∗i (z)Λ∗j (w) =: Λ∗i (z)Λ∗j (w) :

γ− ( wz )j−i =: Λ∗i (z)Λ∗j (w) : γ− ( wz ). γ+ (p wz )j−i−1

At the last line we used the formula γ− (z)/γ+ (pz) = 1. For the final case i > j, we have f1,m ( wz )Λ∗i (z)Λ∗j (w)

=:

Λ∗i (z)Λ∗j (w)

γ+ ( wz )i−j : =: Λ∗i (z)Λ∗j (w) : γ+ ( wz ). γ− (p−1 wz )i−j−1

Thus all the cases are proved. 3.3.5. Proof of (5). This is similary shown as (2) and (3), so we omit the detail. Acknowledgements. S.Y. is supported by JSPS Fellowships for Young Scientists (No.21-2241). References [AKOS]

H. Awata, H. Kubo, S. Odake, J. Shiraishi, Quantum WN algebras and Macdonald polynomials, Comm. Math. Phys. 179 (1996), no. 2, 401–416. [DI] J. Ding, K. Iohara, Generalization of Drinfeld quantum affine algebras, Lett. Math. Phys. 41 (1997), 183–193. [FF] B. Feigin, E. Frenkel, Quantum W-algebras and elliptic algebras, Comm. Math. Phys. 178 (1996), no. 3, 653–678. [FHHSY] B. Feigin, K. Hashizume, A. Hoshino, J. Shiraishi, S. Yanagida, A commutative algebra on degenerate CP1 and Macdonald polynomials, J. Math. Phys. 50 (2009), 095215. [M] I.G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed. Oxford Mathematical Monographs, Oxford University Press, (1995). [SKAO] J. Shiraishi, H. Kubo, H. Awata, S. Odake, A quantum deformation of the Virasoro algebra and the Macdonald symmetric functions, Lett. Math. Phys. 38 (1996), no. 1, 33–51. BF: Landau Institute for Theoretical Physics, Russia, Chernogolovka, 142432, prosp. Akademika Semenova, 1a, Higher School of Economics, Russia, Moscow, 101000, Myasnitskaya ul., 20 and Independent University of Moscow, Russia, Moscow, 119002, Bol’shoi Vlas’evski per., 11 E-mail address: [email protected] AH. Department of Mathematics, Sophia University, Kioicyo, Tokyo, 102-8554, Japan E-mail address: [email protected] JS, JS: Graduate School of Mathematical Sciences, University of Tokyo, Komaba, Tokyo 153-8914, Japan E-mail address: [email protected] E-mail address: [email protected] SY: Kobe University, Department of Mathematics, Rokko, Kobe 657-8501, Japan E-mail address: [email protected]