Plane partitions with a" pit": generating functions and representation ...

arXiv:1512.08779v1 [math.CO] 29 Dec 2015

PLANE PARTITIONS WITH A “PIT”: GENERATING FUNCTIONS AND REPRESENTATION THEORY M. BERSHTEIN, B. FEIGIN, G. MERZON Abstract. We study plane partitions satisfying condition an+1,m+1 = 0 (this condition is called “pit”) and asymptotic conditions along three coordinate axes. We find the formulas for generating function of such plane partitions. Such plane partitions label the basis vectors in certain representations of quantum toroidal gl1 algebra, therefore our formulas can be interpreted as the characters of these representations. The resulting formulas resemble formulas for characters of tensor representations of Lie superalgebra glm|n . We discuss representation theoretic interpretation of our formulas using q-deformed W -algebra glm|n .

1. Introduction In this paper we study certain problems of enumerative combinatorics of 3d Young diagrams, which are motivated by representation theory. It is convenient to identify 3d Young diagrams with plane partitions i.e. collection of nonnegative integers ai,j such that ai,j ≥ ai+1,j , ai,j ≥ ai,j+1 and all but finite number of ai,j equals 0. Later we Pwill also consider more general plane partitions. Denote by |a| = ai,j , i.e. the number of boxes in the corresponding 3d P Young diagram. For any set A of plane partitions define its generating function by a∈A q |a| . Such functions were extensively studied in enumerative combinatorics, for example one of MacMahon’s formulas has the form n−1 X n−1 V (1, q, . . . , q ) , q |a| = q −( 2 ) n (q)∞ {a|an+1,1 =0}

Q

where V (x1 , . . . , xn ) = iA2 >...>An ≥0

(Ai2+1) aν+ρn (q A )aµ+ρn (q −A ) , (q)2n ∞

(1.5)

Note that each summand is a product of two expressions on the right side of (1.4). Since our three formulas are algebraically equivalent it is enough to prove any of them. We give two different combinatorial proofs, one for Theorem 1 and one for Theorem 3. These proofs are simpler then ones of particular cases given in [13], [14]. The first proof is based on a bijection between plane partitions and collections of non crossing paths. The number of such collections is computed using Lindström–Gessel– Viennot lemma [19],[16]. Such proof gives a determinantal expression for χn,m λ,ν,µ (q), see Theorem 1. In the second proof we interpret conditions (1.1),(1.2) as a definition of certain infinite dimensional polyhedron. We compute the generating function of integer points in this polyhedron as a sum of contribution of vertices, using Brion theorem [4]. Such proof gives a “bosonic formula”1 for χn,m λ,ν,µ (q), see Theorem 3. The conditions (1.1),(1.2) appeared in [9] in the context of representation theory of ¨ 1 ). Namely, plane partitions which satisfy these conditions quantum toroidal algebra U~q(gl n,m ¨ 1 ). Therefore χn,m (q) is the label a basis in MacMahon modules Nλ,ν,µ (v) over U~q(gl λ,ν,µ n,m character of the representation Nλ,ν,µ (v). It is natural to ask for representation theoretic interpretation of our character formulas. n,m We claim that there exist resolutions of Nλ,ν,µ (v) such that their Euler characteristics coincide with our character formulas. In such cases we say that resolution is a materialization of character formula. For example BGG resolution [3] is a materialization of Weyl character formula. Zelevinsky constructed complex which is a materialization of Jacobi–Trudi formula for Schur polynomials [28]. Our formulas for functions χn,m λ,ν,µ (q) resemble the formulas for characters of tensor representations of Lie superalgebra glm|n . This similarity can be explained by the fact that the n,m representations Nλ,ν,µ (v) are actually representations of certain q-deformed W -algebra, ˙ n|m ). which we call W~q(gl Such W -algebras appear as follows. There is an easy (but not written in the literature) n,m ¨ 1 ). On such Fock modules fact that Nλ,ν,µ (v) is a subfactor of Fock representation of U~q(gl ¨ 1 ) commutes with certain operators, which are called screening operators. image of U~q (gl Algebras of elements which commute with screening operators are usually called W ˙ n|m ). algebras. The structure of screening operators in our case suggests the name W~q(gl There exists conformal limit q → 1 of screening operators and we denote the algebra ˙ n|m ). For m = 0 this algebra coincides with the which commutes with them by W(gl ˙ n ) [6]. The algebras W(gl ˙ n|1 ) coincide with the Wn(2) algebras introduced in algebra W(gl [15]. We didn’t find reference for generic n, m (note that our W -algebras differ from ones introduced in [17]). 1Usually

a formula is called bosonic if it equals a linear combination of characters of algebra of polynomials. In our case bosonic formula is a combination of terms q ∆ /(q)n+m . ∞

4

M. BERSHTEIN, B. FEIGIN, G. MERZON

Standard statement in the theory of vertex algebras is an equivalence of the abelian categories of certain representations of vertex algebra and certain representations of quantum group. This is a statement similar to Drinfeld–Kohno or Kazhdan–Lusztig theorem. ˙ n|m ) is related to the product of quantum We conjecture that under this equivalence W(gl n,m (v) under groups Uq gln|m ⊗ Uq′ gln ⊗ Uq′′ glm for certain q, q ′ , q ′′ . And representations Nλ,ν,µ (n|m)

(n)

(m)

(n)

(m)

this equivalence goes to the tensor products Lλ ⊗ Lν ⊗ Lµ , where Lν and Lµ are finite dimensional irreducible representations of Uq′ gln and Uq′′ glm correspondingly and (n|m) Lλ is a tensor representation of Uq gln|m . Plan of the paper. In Section 2 we give precise statements of our main results for χn,m λ,ν,µ (q) with necessary notation and comments. The remaining sections 3, 4, 5 are independent of each other. Sections 3 and 4 are devoted to the combinatorial proofs based on Lindström–Gessel–Viennot lemma and Brion theorem correspondingly. Section 5 is devoted to the algebraic discussion, first we give a definition of appropriate W -algebras in terms of screening operators. Conjectural materializations of our character formulas are discussed in subsection 5.5, relation to quantum group Uq gln|m ⊗ Uq′ gln ⊗ Uq′′ glm in subsection 5.7. Acknowledgments. We thank A. Babichenko, E. Gorsky, A. Kirillov, A. Litvinov, I. Makhlin, G. Mutafyan, G. Olshanski, Y. Pugai, A. Sergeev for interest to our work and discussions. The article was prepared within the framework of a subsidy granted to the National Research University Higher School of Economics by the Government of the Russian Federation for the implementation of the Global Competitiveness Program. M. B. acknowledges the financial support of Simons-IUM fellowship, RFBR grant mol a ved 15-32-20974 and Young Russian Math Contest. 2. Main Results 2.1. Due to asymptotic conditions (1.2) aij ≥ νi and aij ≥ µj . In order to define the P grading |a| ∼ aij we need to subtract these asymptotic values νi , µj . We will use the following definition X X |a| = (aij − νi ) + (aij − µj ) (2.1) i−n≤j−m, (i,j)6∈λ

i−n>j−m, (i,j)6∈λ

Note that this definition of grading is not invariant under m, µ ↔ n, ν symmetry. Geometrically the definition (2.1) can be restated as follows. We draw a staircase line from the point (m, n) as on the picture below. This line divides the base of the plane partition a into two parts. We subtract νi from cells in the upper part and µj from cells in the left part, see Fig. 2. Let r = min{t|λn−t ≥ m − t}, 0 ≤ r ≤ min{n, m}. Geometrically r is the number of boxes above the staircase line starting from the point (m, n). In the picture above we have r = 2. Note that this number r has a interpretation in terms of representation theory of gl(m|n) namely r is called the degree of atypicality of the tensor representation of gl(m|n) corresponding to λ.

PLANE PARTITIONS WITH A “PIT”

5

−ν1 −ν1 −ν1 −ν1 −ν2 −ν2 −ν2 −ν2 −ν2 −ν3 −ν3 −ν3 −ν3 −ν3 −µ2 −ν4 −ν4 −ν4 −ν4 −ν4 −µ2 −µ3 −ν5 −ν5 −ν5 −ν5 −µ1 −µ2 −µ3 −µ4 −µ1 −µ2 −µ3 −µ4 −µ1 −µ2 −µ3 −µ4 −µ1 −µ2 −µ3 −µ4

Figure 2. In order to write down the formula for the generating function we parametrize λ by some analogue of Frobenius coordinates. We introduce two partitions π, κ by πi = λi − (m − r) for i = 1, . . . , n − r and κj = λ′j − (n − r) for j = 1, . . . , m − r, where λ′ denotes transpose of the partition λ. We denote components of partitions ν, µ, π, κ shifted by ρ by the corresponding capital Latin letters: Ni = νi + n − i, Mj = µj + m − j, Pi = πi + (n − r) − i, Qj = κj + (m − r) − j. 2.2. In the simplest case n = 1, m = 0 for any asymptotic conditions λ, ν the generating function of partitions χ1,0 ν,∅,λ (q) equals 1/(q)∞ (and similarly for n = 0, m = 1 case). Now consider the n = m = 1 case. If λ 6= ∅ then the plane partitions decompose into two partitions so the generating function equals 1/(q)2∞ . There is an clear bijection between our plane partitions and V -partitions [26] i.w. the N-arrays of integer numbers: a1 a2 a2 . . . a0 , b1 b2 b2 . . . such that a0 ≥ a1 ≥ a2 ≥ . . ., a0 ≥ b1 ≥ b2 ≥ . . ., limi→∞ ai = ν1 , limi→∞ bi = µ1 . The weight of V -partition is defined as X X N= (ai − a) + (bi − b). i≥0

i≥1

Lemma 2.1. The generating function of V -partitions with asymptotic conditions limi→∞ ai = ν1 , limi→∞ bi = µ1 equals i(i+1) ∞ X 2 q di iq , (2.2) R(d; q) := (−1) (q)2∞ i=0

where d = ν1 − µ1 .

6


This lemma can be proved by a kind of inclusion-exclusion argument. For the case ν1 = µ1 see [26, Sec. 2.5]. The general case can be proved in a similar manner. See also [9, Cor. 5.6]. Now we can write down the first formula for χn,m µ,ν,λ (q). Theorem 1. The generating function χn,m µ,ν,λ (q) is equal to the determinant of a block matrix of the size (m + n − r) × (m + n − r)

χn,m µ,ν,λ (q) =

where

∆m,n µ,ν,λ

(−1)mn−r q (q)m+n ∞

∆m,n µ,ν,λ

=

Pm−r j=1

P

 det 

Mj Qj +

(a+1 2 ) q (Nj −Mi )a (−1) q a≥0 q −Nj (Pi +1) 1≤i≤n−r a

1≤i≤m 1≤j≤n

q

−Mi Qj

i=1

1≤i≤m 1≤j≤m−r 

0

1≤j≤n

Pn−r



,

(2.3)

Ni (Pi + 1).

Clearly this formula generalizes previous consideration in the n, m ≤ 1 case, where determinant becomes 1 × 1. One can think that the formula (2.3) is similar to Jacobi– Trudi formula, which expresses generic Schur polynomial sλ in terms of Schur polynomials corresponding to rows (or columns). The Theorem 1 is proven in Section 3. It is natural that the determinant expression for the generating function can be proven using non-intersecting paths and Lindström– Gessel–Viennot lemma. Let us mention two more special cases where we have only one block in the matrix. In the case of m = 0 we have r = 0, π = λ and after a multiplication on (q)n∞ the determinant becomes equal to aν+ρn (q −λ−ρn ). So we get the known formula (1.4). In the case m = n and λ = ∅ the formula (2.3) simplifies to det R(Nj − Mi ; q) . This formula was proven in [14] (following [9]) under the additional assumption that µ = ∅.

2.3. The determinant in formula (2.3) can be calculated. Theorem 2. The generating function χn,m µ,ν,λ (q) is equal to the sum over r-tuples of integer numbers A1 > A2 > . . . > Ar ≥ 0 χn,m µ,ν,λ (q)

= (−1)

m,n r(m+n) ∆µ,ν,λ

q

X

(−1)

A1 >A2 >...>Ar ≥0

r P i=1

Ai

q

r P i=1

(Ai2+1) aN (q A , q −P −1)aM (q −A , q −Q ) , (q)m+n ∞

where aN , aM were defined in formula (1.3) and ∆m,n µ,ν,λ =

Pm−r j=1

Mj Qj +

Pn−r i=1

(2.4) Ni (Pi + 1).

There are two special cases in which the right hand side takes a simpler form. These two special cases of the theorem were known. First, if m = 0 then the formula (2.4) reduces to (1.4). More generally if r = 0, then base of the plane partition decomposes into two connected components and the formula (2.4) become a product aN (q −P −1 )aM (q −Q )/(q)m+n ∞ .


7

-1 0 1

Figure 3. In the second case we take λ = µ = ν = ∅. Then the functions aN and aM reduces to Vandermonde products and we can write (we assume that n ≥ m) χn,m ∅,∅,∅ (q)

X

=

(−1)

A1 >A2 >...>Am ≥0

r P i=1

Ai

q

r P i=1

(Ai2+1) V (q m−n , . . . , q −1 , q Am , . . . , q A1 )V (q −A1 , . . . , q −Am ) . (q)m+n ∞

(2.5) This formula coincides with the [9, Conjecture 5.10] proved in [13, Theorem 1.2] by completely different method. Lemma 2.2. The r.h.s. of (2.3) and (2.4) are equal. Our proof of this lemma is based on a direct caluclation, which we present in Subsection 3.4. 2.4. The sum in formula (2.4) contains zero terms if one of Ai equals to one of Qj . We want to exclude such terms. The following lemma is standard. Lemma 2.3. For any partition λ we have Z = {λ′j − j − (n − m)|j ∈ N}

G {i − λi − (n − m) − 1|i ∈ N}.

Sketch of the proof. The proof is based in the following construction. Rotate Young diagram corresponding to λ by 135◦ and take a projection on OX. On the Fig 3 we give an example for λ = (4, 4, 4, 3, 3, 1). It is easy to see that coordinates of white balls are i − λi − 12 and coordinates of black F ones are λ′j −j + 21 . Therefore Z+ 12 = {λ′j −j + 12 } {i−λi − 12 }. Shifting by −(n−m) − 21 we get the Lemma. Recall that

Qj = κj − j + m − r = λ′j − j − (n − m) for 1 ≤ j ≤ m − r. Note that λ′j − j − (n − m) ≥ 0 if and only if 1 ≤ j ≤ m − r. Since A1 , . . . , Ar ≥ 0 and Ai 6= Qj then {Ai } should be a subset in the complement set {i − λi − (n − m) − 1|j ∈ N}. Note that i − λi − (n − m) − 1 ≥ 0 if and only if i > n − r. Therefore the non zero terms correspond to subsets {Ar , . . . , A1 } ⊂ {−Li |i > n − r}, where Li = λi − i + n − m + 1. Rewriting the determinants in (2.4) as a sum over permutations we get.

8


Theorem 3. The generating function χn,m µ,ν,λ (q) is equal to the sum χn,m µ,ν,λ (q)

= (−1)

r(m+n)

X

(−1)

|σ|+|τ |+

r P i=1

(σ,τ,A)∈Θ

Ai

σ,τ,A

(µ,ν,λ) q∆ , n+m (q)∞

(2.6)

where (σ, τ, A) ∈ Θ ⇔ σ ∈ Sn , τ ∈ Sm , Ai = −Lsi , for s1 > · · · > sr > n − r, and σ,τ,A

∆

(µ, ν, λ) =

r X

Ai

i=1

Ai + 1 + Nσ(i) − Mτ (i) 2 −

n X

(Pi−r + 1)(Nσ(i) − Ni ) −

i=r+1

m X

Qi−r (Mτ (i) − Mi ).

i=r+1

This theorem will be proven in Section 4 (in the case ν1 > . . . > νn > µ1 > . . . > µm ). In this proof we consider inequalities ai,j ≥ ai+1,j , ai,j ≥ ai,j+1 and conditions (1.2) as a definition of a polyhedron (infinite dimensional) and the generating function χn,m µ,ν,λ (q) as a sum over integer points in the polyhedron. This sum is calculated using Brion theorem [4]. Each term in (2.6) corresponds to a vertex contribution in Brion theorem. 3. Lattice paths 3.1. Let G be an oriented locally finite graph with set of vertices V and set of edges E. We also assume that G have no oriented cycles. For any edge e ∈ E we assign a weight w(e), for any Q path p = (e1 , e2 , . . . , en ) we define the weight as a product of the edge weights w(p) = w(ei ). P For any two vertices s, t we denote P (s → t) = p w(p), where summation goes over all paths from s to t. For any sets of n source vertices P S = {s1 , . . . , sn } and n target vertices T = {t1 , . . . , tn } we denote P (S → T ) = p1 ,...,pn w(p1 ) · . . . · w(pn ), where summation goes over all sets of path such that pi goes from si to ti . Clearly P (S → T ) = P (s1 → t1 ) · . . . · P (sn → tn ).P By Pnc (S → T ) we denote the sum p1 ,...,pn w(p1) · . . . · w(pn ) where set of paths is assumed to be without crossings. The Lindström-Gessel-Viennot lemma provides an efficient way to find Pnc (S → T ). Lemma (Lindström-Gessel-Viennot; [19],[16]). For oriented graph G as above and any sets of sources and targets S = {s1 , . . . , sn }, T = {t1 , . . . , tn } we have X n (−1)|σ| Pnc (S → σ(T )) = det P (si → tj ) i,j=1 σ∈Sn

In most examples (and in all examples in this paper) Pnc (S → σ(T )) 6= 0 for only one permutation σ. In this case n Pnc (S → σ(T )) = (−1)|σ| det P (si → tj ) i,j=1 .


9

3.2. In this paper we use graph G with vertices (a + 21 , b), where a, b ∈ Z, b ≥ 0. There are two types of edges namely the horizontal ones (a+ 21 , b) → (a+ 23 , b) (→ denotes orientation) and vertical ones (a + 12 , b) → (a + 12 , b + 1) for a < 0 and (a + 21 , b) ← (a + 21 , b + 1) for a ≥ 0. The weight of a vertical edge is 1, the weight of a horizontal edge on the line y = b is q b . Note that the number of paths from s = ( 12 , b) to t = ( 21 + a, 0), a, b ≥ 0 is equal to binomial coefficient a+b . The number of paths counted with weights is equal to the b q-binomial coefficient P (s → t) = a+b . b q We will use “infinitely remote” source and target vertices, see an example in Fig. 4. We say that the path starts at point (−∞, b) if the path contains all sufficiently left edges on the horizontal line y = b. Similarly we define paths which starts at point (a, +∞) or goes to the point (+∞, b) or (a, +∞). For example the paths from the point s = (−∞, 0) to t = (− 12 , +∞) are in one to one correspondence with Young diagrams. And in this case P (s → t) is equal to the generating function of Young diagrams 1/(q)∞ . For the ‘infinitely remote” source and target vertices we need to define the weight of the path. The problem happens for vertices (−∞, b) since their paths contain infinitely many horizontal edges on the line y = b and therefore the weight of these paths are not defined. We divide by q b the weight of each horizontal edge (of such paths) over the point (i, 0), i < 0. Clearly there is no more then one such edge, if there is none we just divide the weight of the path by q b . Informally speaking we assign the weight q −b(∞/2−1/2) to the vertex (−∞, b). For example for s = (−∞, b), t = (−a − 21 , +∞), a, b ≥ 0 we have P (s → t) = q −ab /(q)∞ . For the (+∞, b) we divide by q b the weight of each horizontal edge (of path to the (+∞, b)) over the point (i, 0), i ≥ 0. Informally speaking we assign the weight q −b(∞/2+1/2) to the vertex (−∞, b). For example for s = (a + 12 , +∞), t = (+∞, b), a, b ≥ 0 we have P (s → t) = q −(a+1)b /(q)∞ . Now we prove the formula (1.4) for the number of plane partitions with n rows and asymptotic conditions. Proposition 3.1. Generating function of plane partition aij , such that 1 ≤ i ≤ n, j ∈ N, (i, j) 6∈ λ, limi→∞ aij = νi has the form

χn,0 ∅,ν,λ (q)

=

q

Pn

i=1 (λi +n−i)(νi +n−i)

(q)n∞

aν+ρn (q −λ−ρn ),

Proof. There is natural bijection between such plane partitions and collections of non intersecting paths from S = s1 , . . . , sn , si = (λi + n − i + 12 , +∞) to T = t1 , . . . , tn , ti = (+∞, νi + n − i). The first row of the plane partition encodes the path from s1 to t1 , the second row of the plane partition encodes the path from s2 to t2 and so on. The coordinates of the sources and targets are specified in such way that plane partition condition ai,j ≥ ai+1,j is equivalent to the non intersection of paths. In the Fig. 4 we give an example, where n = 3, λ = (2, 1, 1), ν = (3, 1, 1).

10


s3 s2

∞ ∞ 3 3 3 3 3 3 3 ... 3 ∞ 3 3 2 1 1 1 1 1 ... 1 ∞ 3 3 1 1 1 1 1 1 ... 1

←→

s1

t1

t2 t3

Figure 4. As was noted before we have P (si → tj ) = q −(λi +n−i+1)(νj +n−j) /(q)∞ . Therefore using Lindström-Gessel-Viennot lemma we get −(λ +n−i+1)(ν +n−j) j q i P (S → T ) = det . (q)∞

Note that the function P (S → T ) differs from χn,0 ∅,ν,λ (q) by certain power of q since on grading on paths differs slightly from the definition (2.1). In particular χn,0 ∅,ν,λ (q) has leadP − i (λi +n−i+1)(νi +n−i) ing term 1 but P (S → T ) has leadign term q . Multiplying P (S → T ) P by q i (λi +n−i+1)(νi +n−i) we get Proposition 3.1.

3.3. We had not discussed one type of pathes between “infinitely remote” vertices. Namely let s = (−∞, b), t = (+∞, a). Then the paths from s to t are in one to one correspondence with V -partitions with asymptotic conditions limi→∞ ai = a, limi→∞ bi = b, see Fig. 5.

4 4 2 2 ... 5 3 3 1 1 ...

←→ t

s

Figure 5. Recall that the generating function of V partitions was given in Lemma 2.1 and equals R(a − b; q). Now we are ready to prove Theorem 1. Proof of Theorem 1. First we use a one to one correspondence between plane partitions satisfying (1.2),(1.1) and certain lattice paths. We decompose the base of plane partition into r infinite hooks, m − r infinite columns and n − r infinite rows.


11

We set the sources and targets to the points  for 1 ≤i ≤ m,  (−∞, Mi ) , si = 1  (Pi−m + , +∞) for m + 1 ≤i ≤ m + n − r 2  for 1 ≤j ≤ n,  (+∞, Nj ) tj = . 1  (−Qj−n − , +∞) for n + 1 ≤j ≤ m + n − r 2 We illustrate correspondence in the Fig. 6, where we have n = 3, m = 2, λ = (2, 1, 1), ν = (3, 1, 1), µ = (2, 0). By previous definitions r = 1, π = (1, 0), κ = (1), N1 = 5, N2 = 2, N3 = 1, M1 = 3, M2 = 0, P1 = 2, P2 = 0, Q1 = 1. t4

s4

s3

∞ ∞ 3 3 3 3 3 3 3 ... 3 ∞ 3 3 2 1 1 1 1 1 ... 1 ∞ 3 3 1 1 1 1 1 1 ... 1 4 2 ←→ 3 1 2 0 s1 2 0 ...... 2 0

t1

t2 t3

s2

Figure 6. Due to our order of si and tj non intersecting paths correspond to the permutation s1 s2 . . . sm−r sm−r+1 . . . sm sm+1 . . . sm+n−r−1 sm+n−r tn+1 tn+2 . . . tm+n−r tn−r+1 . . . tn t1 . . . tn−r−1 tn−r

We denote this permutation of indexes 1, . . . , m + n − r by σm,n,r . So we proved that χn,m µ,ν,λ (q) = P (S → σm,n,r (T )).

The value P (S → σm,n,r (T )) we compute from the Lindström-Gessel-Viennot lemma. The number of inversion in the permutation σm,n,r is equal to mn − r 2 . It remains to write P (si → tj ), which have been actually found above  R(Nj − Mi ; q) for 1 ≤ i ≤ m, 1 ≤ j ≤ n,      q −Mi Qj−n /(q)∞ for 1 ≤ i ≤ m, n+1≤ j ≤m+n−r P (si → tj ) = . −Nj (Pi−m +1)  q /(q) for m + 1 ≤i ≤ m + n − r, 1 ≤ j ≤ n  ∞   0 for m + 1 ≤i ≤ m + n − r, n + 1 ≤ j ≤ m + n − r Pm−r

Combining all together we obtain Theorem 1. As above, additional factor q j=1 Mj Qj + comes from the difference between definition of grading in terms of paths and (2.1).

Pn−r i=1

Ni (Pi +1)

12


3.4. In this Subsection we prove Lemma 2.2. Proof. We want to calculate the determinant of the matrix  P a+1 −Mi Qj a ( 2 ) (Nj −Mi )a q q 1≤i≤m a≥0 (−1) q 1≤i≤m  1≤j≤m−r  1≤j≤n M= . 0 q −Nj (Pi +1) 1≤i≤n−r 1≤j≤n

We decompose this matrix as a product of two (infinite) matrices   a+1 0 (−1)a q ( 2 )−aMi 1≤i≤m, aN q j a∈Z, a≥0  M = C 1≤j≤n (δ−a−1,Pi )1≤i≤n−r, 0

(−1)Qj δa,Qj

aA2 >...>Ar ≥0

Therefore we proved that (2.3) is equal to (2.4).

4. Integer Points in Polyhedra 4.1. In this section we give a combinatorial proof of Theorem 3 in the case ν1 > . . . > νn > µ1 > . . . > µm .

(4.1)

This proof is based on Brion’s theorem which we briefly recall. Let P ⊂ RN be a convex polyhedron i.e. an intersection of finite number of half-spaces. Note that P can be unbounded. For simplicity we assume below that vertices of P have integer coordinates and edges have rational directions. For point p = (p1 , . . . , pN ) ∈ ZN by tp we denote tp11 · · · tpNN . Define the characteristic function of P by the formula X tp . S(P ) = p∈P ∩Zn

±1 In this definition S(P ) is a formal series, S(P ) ∈ Z[[t±1 1 , . . . tN ]]. It can be proven that ±1 ±1 there exist two Laurent polynomials f, g ∈ Z[t1 , . . . tN ] such that f S(P ) = g. We denote S(P ) = f /g ∈ Q(t1 , . . . , tn ). Clearly S(P ) does not depend on the particular choice of ±1 f, g ∈ Z[t±1 1 , . . . tN ] For any vertex v ∈ P , we denote by Kv its cone i.e. the intersection of half-spaces corresponding to the facets (maximal proper faces) of P containing v. The next theorem is called Brion theorem, standard references for this theorem are [4], [20], [21], [18], for a clear introduction see e.g. [1].

,


13

Theorem (Brion). For any convex polyhedron P with integer vertices and rational directions of edges we have X S(P ) = S(Kv ). v

n,m Plane partitions satisfying (1.1) and (1.2) are integer points of the polyhedron Pµ,ν,λ defined as follows  ti,j ≥ νi ≥ 0   ti,j ≥ ti,j+1 n,m ti,j ≥ ti+1,j ti,j ≥ µj ≥ 0 , (i, j) ∈ N2 \ λ. Pµ,ν,λ : (4.2)  t n+1,m+1 = 0

Therefore the functions χn,m µ,ν,λ (q) can be computed using Brion theorem. Two remarks are in order. First, we state Brion theorem for finite dimensional polyn,m n,m hedra but Pµ,ν,λ is infinite dimensional. Therefore we start from finitization of Pµ,ν,λ , i.e. n,m,(H) for H ∈ N we consider polyhedron Pµ,ν,λ defined as  ti,H = νi ≥ 0   ti,j ≥ ti,j+1 n,m,(H) ti,j ≥ ti+1,j tH,j = µj ≥ 0 , (i, j) ∈ {1, . . . , H}2 \ λ. Pµ,ν,λ : (4.3)  t n+1,m+1 = 0 Then we take the limit H → ∞. Second, we need specialization of the function S(P ) in which ti,j → q. We denote by Sq (P ) ∈ Q(q) the function obtained by composition of S and this specialization 2 The limit (H) n,m,(H) (H) q −∆ Sq (Pµ,ν,λ ) coincides with χn,m emerge due to different µ,ν,λ (q). Here numbers ∆ definitions of grading, see below. It will be convenient to start from specialization ti,j → xj−i+1 /xj−i . We denote by Sx (P ) ∈ Q({xi }) composition of S and this specialization. Then we can set xi → q i and get Sq (P ).

4.2. We explain main ideas in the case m = 0. As the result we get new proof of (1.4). n,0,(H) We start from a description of vertices of the polyhedron P∅,ν,λ . Since tn+1,1 = 0 one can think that indices of coordinates ti,j satisfy 1 ≤ i ≤ n, 1 ≤ j ≤ H, (i, j) 6∈ λ. Any face of our polyhedron is defined by (4.3) where some of inequalities become equalities. For any face we construct graph Γ with vertices (i, j) ∈ (H n ) − λ. Two vertices (i, j) and (i′ , j ′) are connected by an edge iff ti,j = ti′ ,j ′ for all points of the face and boxes (i, j) and (i′ , j ′) have a common side. There exist at least n connected components in Γ since ti,H = νi and νi > νj for i > j (due to (4.1)). Vertices of our polyhedron are faces of maximal codimension i.e. corresponding to graphs having exactly n connected components. See an example in Fig. 7. Denote by Γv graph corresponding of vertex v. Denote by Ks connected components of Γv . Each Ks is a skew Young diagram. Denote by Ks,v projection of the Q cone Kv on the subspace with coordinated ti,j for (i, j) ∈ Ki . Then we have S(Kv ) = S(Ks,v ). 2In

general, such specialization of a rational function might not be well defined, but in our case n,m,(H) n,m,(H) Sq (Pµ,ν,λ ) and Sq (Kv ), where v is vertex of Pµ,ν,λ are well defined.

14


ν1 ν2 ν3 ν4

Figure 7. Situation simplifies since for many vertices Sq (Kv ) vanishes due to the following result. Proposition 4.1 ([22, Theorem 3.1]). If connected component Ks has cycles, then Sx (Ks,v ) is equal to 0. Therefore by Brion theorem we have n,0,(H)

Sq (P∅,ν,λ ) =

X

Sq (Kv ),

(4.4)

v

where summation goes over vertices v such that corresponding graphs Γv are acyclyc. Recall that a skew Young diagram α − β is called ribbon if it is connected and contains n,0,(H) no 2 × 2 block of squares. Due to Proposition 4.1, vertices of P∅,ν,λ with nonzero contribution correspond to decompositions of skew diagram (H n ) − λ into n ribbons such that boxes (H, i) belong to different ribbons.3 The following lemma is standard. Lemma 4.1. If α − β is a ribbon then there exist j, k ∈ N such that the set {αi − i} is obtained from the set {βi − i} by replacement of βj − j by βj − j + k The set {λj − j} is called the set of shifted parts of partition λ. Recall useful geometric interpretation of set {j − λj − 21 } is a set of the coordinates of white balls in the Fig. 3. The corresponding sets for partitions (H n ) and λ differs in first n numbers. Therefore to any decomposition of skew diagram (H n ) − λ into n ribbons we assign a permutation σ ∈ Sn such that our ribbons shift λi − i to H − σ(i). Since boxes (H, i) correspond to different ribbons the order of ribbons is well defined and this assignment is one to one correspondence. For example the graph Γv in the Fig. 8 corresponds to the permuta 1234 . tion 1432 We denote by vσ the acyclic vertex corresponding to σ ∈ Sn . n,0,(H)

Lemma 4.2. For vertex vσ of polyhedron P∅,ν,λ |σ| ∆σ,(H) (λ,ν)

Sq (Kvσ ) = (−1) q

/

we have

n Y

(q)H−σ(i)−λi +i−1 ,

i=1

Qk

− q s ) and n n X X σ,(H) 2 ∆ (λ, ν) = (Hνi − iλi − iνi + i ) − (λi − i)(νσ(i) − σ(i)).

where (q)k =

s=1 (1

i=1

3One

can compare this to Murnaghan–Nakayama rule.

i=1

(4.5)


15

j − i = λ1 − 0

ν1

j − i = λ2 − 1 j − i = λ3 − 2

ν2

j − i = λ4 − 3

ν3 ν4

Figure 8. The proof is similar to the one in [22, Prop. 3.4]. Proof. Let Ki be a ribbon which shifts λi − i to H − σ(i). The set {Ki } is a set of all connected components of graph Γvσ . First, we compute the contribution of the corresponding cone S(Kvσ ,i ). Denote by h number of boxes in ribbon Ki , clearly h = H − σ(i) − λi + i. We number these boxes by 1, . . . h from bottom-left corner to top-right corner. In order to simplify notations we denote the corresponding coordinates tij by t1 , . . . , th . In the vertex vi of cone Kvσ ,i all this coordinates equal to νσ(i) , therefore S(tvi ) = q νσ(i) h . The cone Kvσ ,i is simple. Its edges generated by vectors e1 , . . . , eh−1 , where the vector es equals ±(1, . . . , 1, 0, . . . , 0), with s nonzero coordinates. The sign “±” is equal to “+” if side between the boxes s and s + 1 is vertical and is equal to “−” otherwise. See an example in Fig. 9 +1

−1

e1

−1

+1

+1

+1 +1

−1 −1 −1

−1 −1 −1

+1 +1 +1

+1

+1

−1

−1

+1

e2

e3

e4

e5

e6

Figure 9. Q ±s Therefore Sq (Kvσ ,i ) = q νσ(i) h / h−1 s=1 (1 − q ), where signs “±” were specified above. Now we want to express this product in more explicit terms. We draw lines given by equations j = i + c, where c ∈ Z. Such lines go through centers of boxes in our skew diagram (H n ) − λ. Such line intersects ribbon Ki in one box if λi − i < c ≤ H − σ(i) and do not intersects otherwise. If side between the boxes s and s + 1 in ribbon Ki is horizontal then the line i − j = c passing through center of s + 1 box intersect the ribbon below Ki . If it does not pass through the center of the first box in that ribbon then it intersects the ribbon below and so on. Therefore, this line intersects the first box of certain ribbon, say Kj . Then c = λj − j + 1 and s = λj − j − λi + i. In such case σ(i) < σ(j) since Kj is below Ki , but i > j since s > 0. And, conversely for any such j we will have edge es with “−” sign.

16


Rewriting factors 1/(1 − q −s ) as −q s /(1 − q s ) and using formula for h we have ,H−σ(i)−λi +i−1 P νσ(i) (H−σ(i)−λi +i)+ (λj −j−λi −i) Y jσ(i) Sq (Kvσ ,i ) = (−1)|{j:jσ(i)}| q (1 − q s ) s=1

Qn

Now we can find Sq (Kvσ ) = i=1 Sq (Kvσ ,i ). Using algebraic identities n X X λj − j − λi + i = (λi − i)(σ(i) − i) jσ(i)

and

n X

(4.6)

i=1

νσ(i) (H − σ(i) − λi + i) +

n X

(λi − i)(σ(i) − i) = ∆σ,(H) (λ, ν)

i=1

i=1

we get (4.5).

n,0,(H)

Now we can find Sq (P∅,ν,λ ) using specialization of Brion theorem (4.4) n,0,(H)

Sq (P∅,ν,λ ) =

X

(−1)|σ| q ∆

σ,(H) (λ,ν)

σ∈Sn n,0,(H) P∅,ν,λ ∆(H)

/

n Y (q)H−σ(i)−λi +i−1 . i=1

P

Here we count integer points in with the weight q tij , which differs from the P weight defined in formula (2.1) by q , where ∆(H) = ni=1 νi (H − λi ). Using identity n n X X σ,(H) (H) ∆ (λ, ν) − ∆ = (νi + n − i)(λi + n − i) − (λi + n − i)(νσ(i) + n − σ(i)) i=1

we see that limit limH→∞ q

−∆(H)

i=1

n,m,(H) Sq (Pµ,ν,λ )

coincides with the right side of (1.4)

4.3. We will prove Theorem 3 in the case (4.1). We believe that our methods can be generalized for any other order of νi , µj . Proof. As in before for any vertex v we construct graph Γv . It follows from Proposition 4.1 that vertices with non zero contribution in Sq corresponds to decomposition of skew diagram (H n , mH−n ) − λ into m + n ribbons (connected components of Γv , where n contain boxes (i, H) 1 ≤ i ≤ n and m contain boxes (H, j) 1 ≤ j ≤ H. For any partition α we consider the set of shifted parts {αi − i + n − m + 1}. Note that this set differs from one used in Lemma 4.1 by n − m + 1. We recall notation from Section 2: {Li = λi − i + n − m + 1} and Li = Pi + 1, for 1 ≤ i ≤ n − r, Li ≤ 0 for i > n − r. The set of shifted parts for (H n , mH−n ) equals {H + n−m−1, . . . , H −m+ 1, 0, . . . , −H + n+ 1, −H + n−m, . . .}. Due to Lemma 4.1 addition of n ribbons containing boxes (i, H) should replace n numbers B1 > B2 > . . . > Bn , Bi ∈ {Ls |s ∈ N} by numbers H + n − m − i. The numbers Pi + 1 should belong to the set {Bi }, therefore (B1 , B2 , . . . , Bn ) = (P1 + 1 . . . , Pn−r + 1, −Ar , . . . , −A1 ),

(4.7)

where Ai ∈ {−Ls |s > n − r}. For any vertex we assign permutation σ ∈ Sn such that our n ribbons replace Bi by H +n−m+1−σ(i). These data σ ∈ Sn and Ai ∈ {−Ls |s > n−r} encodes n ribbons containing (i, H).


17

Due to Lemma 2.3 the set of shifted parts {Li } have m − r negative holes (missing negative numbers) in integers −Qj . After adding first n ribbons we have m holes in integers −Cj where (C1 , C2 , . . . , Cm ) = (Q1 . . . , Qm−r , A1 , . . . , Ar ).

(4.8)

The set of shifted parts for (H n , mH−n ) have holes in integers −H + n + j, 1 ≤ j ≤ m. Therefore the m ribbons containing boxes (H, j) replace integers −H + n + τ (j) by −Cj . We denote by vσ,τ,A corresponding vertex, it is easy to see that data determine these 123 1234 , , τ = vertex uniquely. In the example in Fig. 10 we have σ = 132 1342 A1 = 2, A2 = 0. j − i + n − m = P1 + 1

ν1

j − i + n − m = P2 + 1

ν2

j − i + n − m = −A2

ν3 ν4

j − i + n − m = −A1

ν5 j − i + n − m = −A2 − 1

ν6 ν7

j − i + n − m = −A1 − 1

ν8 ν9

j − i + n − m = −Q1 − 1

ν10

µ1 µ2 µ3 µ4 µ5 µ6 µ7 µ8 µ9 µ10

Figure 10. Therefore by Brion theorem we have n,m,(H)

Sq (Pµ,ν,λ

)=

X

Sq (Kvσ,τ,A ),

(4.9)

σ,τ,A

where σ ∈ Sn , τ ∈ Sm , Ai = −Lsi , for s1 > · · · > sr > n − r and Ai < H − n. n,m,(H)

Lemma 4.3. For vertex vσ,τ,A of polyhedron Pµ,ν,λ P

we have σ,τ,A,(H)

(µ,ν.λ) (−1)|σ|+|τ |+ Ai −i+1 q ∆ Qm , Sq (Kvσ,τ,A ) = Qn i=1 (q)H−σ(i)+n−m−Bi j=1 (q)H−τ (i)+m−n−Ci −1

(4.10)

18


where σ,τ,A,(H)

∆

(µ, ν, λ) =

r X

Ai

i=1

+

m−r X

X n−r Ai + 1 − Nσ(n−i+1) + Mτ (m−r+i) + (Pi +1)(−Nσ(i) +n−i)+ 2 i=1

Qj (−Mτ (j) + m − j) +

j=1

n X

νi (H + n − m − i + 1) +

i=1

n X

µj (H + n − m − j)

j=1

(4.11)

and Ni = νi + n − i, Mj = µj + m − j. Proof. The proof of this lemma is analogous to the one of Lemma 4.2. In the denominator we have product (q)h−1 where h is the length of the ribbon. The power of q in the numerator made from two summands. The first one n m X X νσ(i) (H − σ(i) + (n − m) + 1 − Bi ) + µτ (j) (H − τ (h) + m − n − Cj ) j=1

i=1

is the sum of weights of vertices vi in cones Kvσ,τ,A,i corresponding to ribbons. The second one X m n r r X X X Ai + 1 Cj (τ (j) − j) Bi (σ(i) − i) + + Ai (2i − r − 1) + 2 j=1 i=1 i=1 i=1

comes from rewriting 1/(1−q −s ) as −q s /(1−q s). Here we also used identity (4.6). Putting all things together and using (4.7), (4.8) we get (4.11). −∆ Now we find χn,m µ,ν,λ (q) as the limit limH→∞ q (H)

∆

=

n−r X

νi (H − Pi − i + n − m) +

i=1

m−r X

(H)

n,m,(H)

Sq (Pµ,ν,λ

), where

µj (H − Qj − j + m − n)

j=1

+

n X

i=n−r+1

νi (H − i + n − m + 1) +

m X

µj (H − j + m − n).

j=m−r+1

It is easy to see that ∆σ,τ,A,(H) (µ, ν, λ) − ∆(H) = ∆σ˜ ,˜τ ,A (µ, ν, λ) for 1 ... r r + 1 ... n 1 ... r r + 1 ... m σ ˜ = σ◦ , τ˜ = τ ◦ n ... n −r + 1 1 ... n− r m− r + 1 ... m 1 ... m− r and we get formula (2.6).

5. Algebras, representations and resolutions ¨ 1 ) one can use [10, Sec 2] but our 5.1. For the reference of quantum toroidal algebra U~q(gl 4 notation slightly differs from loc. cit. Fix complex numbers ǫi , where i = 1, 2, 3 and should be viewed as a mod 3 residues. We assume that ǫ1 + ǫ2 + ǫ3 = 0. Denote qi = eǫi , ~q = (q1 , q2 , q3 ). We assume further that ¨ ) other currently there is no standard convention to the notations, even for the algebra besides Uq~(gl 1 ¨q1 ,q2 ,q3 (gl1 ) are also used in the literature. names E, E1 , SH, U 4


19

q1 , q2 , q3 are generic, i.e. for integers l, m, n ∈ Z, q1l q2m q3n = 1 holds only if l = m = n. We set 3 3 3 Y Y X X r/2 −r/2 g(z, w) = (z − qi w), κr = (qi − qi ) = (qir − qi−r ), δ(z) = zm. i=1

i=1

i=1

m∈Z

¨ 1 ) is generated by Em , Fm , Hr where m, r ∈ Z, r 6= 0 and invertible The algebra U~q (gl central elements K0 , C, C ⊥ . In order to write relations we form the currents (generating functions of operators) ! X κr X X Fm z −m , K ± (z) = K0 (C ⊥ )±1 exp ∓ H±r z ∓r . Em z −m , F (z) = E(z) = r r>0 m∈Z m∈Z

The relations have form

g(z, w)E(z)E(w) + g(w, z)E(w)E(z) = 0, K ± (z)K ± (w) = K ± (w)K ± (z),

g(w, z)F (z)F (w) + g(z, w)F (w)F (z) = 0,

g(C −1 z, w) − g(w, C −1z) + K (z)K + (w) = K (w)K − (z), g(Cz, w) g(w, Cz)

g(z, w)K ± (C (−1∓1)/2 z)E(w) + g(w, z)E(w)K ± (C (−1∓1)/2 z) = 0, g(w, z)K ± (C (−1±1)/2 z)F (w) + g(z, w)F (w)K ±(C (−1±1)/2 z) = 0 , 1 Cz − Cw + [E(z), F (w)] = (δ K (w) − δ K (z)), κ1 z w Sym z2 z3−1 [F (z1 ), [F (z2 ), F (z3 )]] = 0. Sym z2 z3−1 [E(z1 ), [E(z2 ), E(z3 )]] = 0, z1 ,z2 ,z3

z1 ,z2 ,z3

¨ 1 ) by autoThere exists an action of the group SL(2, Z) on the toroidal algebra U~q(gl ⊥ ⊥ ⊥ morphisms. We denote by Em , Fm , Hr images of generators Em , Fm , Hr after rotation 0 1 . −1 0 Denote by d the grading operator [d, C] = [D, C ⊥ ] = [d, K0 ] = 0. ¨ 1 ). Let V be a Sometimes it is convenient to consider d as an additional generator U~q(gl ¨ 1 ) such that one can define action of d on the space V with finite representation of U~q(gl dimensional eigenspaces. By the character χ(V ) denote the trace of operator D = q −d where q is a formal variable. ¨ 1 ) has the following formal coproduct5 The algebra U~q(gl [d, Em ] = mEm ,

[d, Fm ] = mFm ,

[d, Hr ] = dHr ,

∆(Hr ) = Hr ⊗ 1 + C −n ⊗ Hr , ∆(H−r ) = H−r ⊗ C r + 1 ⊗ H−r , ∆(E(z)) = E C2−1 z ⊗ K + C2−1 z + 1 ⊗ E (z) , ∆(F (z)) = F (z) ⊗ 1 + K − C1−1 z ⊗ F (C1−1 z),

∆(X) = X ⊗ X, for X = K0 , C, C ⊥ , D, where C1 = C ⊗ 1, C2 = 1 ⊗ C. 5note

⊥ ⊥ that our Em , Fm , Hr are called e⊥ m , fm , hr in [10] (up to rescaling of hr )

r>0 (5.1)

20


¨ 1 ) considered in this paper we have C ⊥ = K0 = 1. In all representations of U~q(gl ¨ 1 ). The In the paper [9] authors defined the MacMahon modules of the algebra U~q (gl MacMahon modules depend on the parameters v, c and three partitions ν, µ, λ, where c is a value of central element C. These modules are denoted by Mµ,ν,λ (v, c). This module has the basis |ai, where a is a plane partition which satisfy condition (1.2). The action of d on Mµ,ν,λ(v, c) is defined by d|ai = |a||ai. Therefore the character χ(Mµ,ν,λ (v, c)) is equal to the generating function of plane partitions satisfying (1.2). The modules Mλ,µ,ν (v) were originally defined by the explicit formulas for the action ⊥ of “rotated” generators Em , Fm⊥ , Hr⊥ in the basis labeled by plane partitions. For example ⊥,± the action of K (z) have the form 1 − c2 v/z Y ψi,j,k (v/z)|ai (5.2) K ⊥,± (z)|ai = 1 − v/z (i,j,k)∈a

where

(1 − q1i−1 q2j q3k v/z)(1 − q1i q2j−1q3k v/z)(1 − q1i q2j q3k−1v/z) . (1 − q1i+1 q2j q3k v/z)(1 − q1i q2j+1q3k v/z)(1 − q1i q2j q3k+1v/z) Notation (i, j, k) ∈ a means that (i, j, k) belongs to the corresponding 3d Young diagram, see Fig. 1. It is easy to see that the product in right side of (5.2) after cancellation of common factors becomes finite. The highest weight of Mµ,ν,λ (v, c) is given by the formula (5.2) applied for “minimal” plane partition a satisfying conditions (1.2). For the generic values c, v, q1 , q2 , q3 the module Mµ,ν,λ(v, c) is irreducible. But for n/2 m/2 c = q1 q2 (and generic v, q1 , q2 , q3 ) this module has one singular vector. The quotient by the submodule generated by this vector is irreducible. This quotient is denoted by n,m Nµ,ν.λ (v) and has the basis |ai where a is a plane partition, satisfying both conditions (1.2) and (1.1). Recall that partitions λ, µ, ν satisfy l(ν) ≤ n, l(µ) ≤ m, and λn+1 < m + 1. n,m Therefore the character χ(Nµ,ν,λ (v)) is equal to the generating function χn,m µ,ν,λ (q) defined in the introduction. This is the representation theoretic interpretation of the left side of (2.3), (2.4), (2.6). Now we will discuss the representation theoretic interpretation of the right side. ψi,j,k (v/z) =

5.2. It is difficult to write down the explicit action of generators En , Fn , Hm in modules n,m Nλ,µ,ν (v). Now we recall construction of another class of modules, namely the Fock modules and intertwining operators between them, which are called screening operators. n,m Then we sketch construction of MacMahon modules Nλ,µ,ν (v) in these terms. ¨ 1 ) comes from the fact that the representation The name of Fock modules over U~q(gl space is identified with the Fock module over some Heiseinberg algebra. In these representations the currents E(z), F (z), K ± (z) are given in terms of the Heisenberg algebra (as combination of vertex operators). (i) We start from the basic Fock modules Fu , where u = ep , p ∈ C and i = 1, 2, 3. The 1/2 central charges of these representations are C = qi , K0 = C ⊥ = 1. This representation space is a module over Heisenberg algebra with generators an , n ∈ Z and relations r/2

[ar , as ] = r

(qi

−r/2 3

− qi −κr

)

δr+s,0 .

(5.3)

PLANE PARTITIONS WITH A “PIT” (i)

21

(i)

We denote by vu the highest weight vector of Fu such that ar vu(i) = 0, for r > 0; (i)

a0 vu(i) = −

ǫ2i p (i) v . ǫ1 ǫ2 ǫ3 u

(i)

Then the representation ρu in the space Fu is defined by the formulae ! ! ∞ ∞ −r/2 X X qi κr κr u(1 − qi ) r −r (i) exp , a−r z exp ar z ρu (E(z)) = r/2 −r/2 r/2 −r/2 κ1 − qi )2 − qi )2 r=1 r(qi r=1 r(qi ! ! ∞ ∞ r/2 −1 −1 X X u (1 − q ) −κ −q κ r r i i ρ(i) exp a z r exp a z −r , u (F (z)) = r/2 −r/2 2 −r r/2 −r/2 2 r κ1 r(q − q ) r(q − q ) r=1 r=1 i i i i a r 1/2 (i) ⊥ , ρ(i) ρu(i) (C) = qi , ρ(i) u (K0 ) = ρu (C ) = 1, u (Hr ) = r/2 −r/2 qi − qi (p + ǫi )3 − p3 (i) (i) ρ(i) (d)v = ∆ (p)v , ∆ (p) = . (5.4) i i u u u 6ǫ1 ǫ2 ǫ3 Note that generally speaking the operators a0 and d can act on the highest weight (i) vector vu by any number. Our choice is convenient for the formulas below, for example the screening operators will commute with d due to our choice. 3 b by the relation [an , Q] b = −ǫi δn,0 . The exponent exQb acts We introduce operators Q ǫ1 ǫ2 ǫ3 (i)

(i)

(i)

(i)

from Fu to Fqix u and maps highest weight vector vu to the vector vqix u . (i)

The modules Fu are irreducible. In terms of rotated generators its highest weight has the form 1 − qi u/z (i) K ⊥,± (z)vu(i) = v (5.5) 1 − u/z u (1)

In particular the highest weight of Fock module Fu coincides with the highest weight of 1/2 Macmahon modules M∅,{ν1},{λ1 } (v) for c = q1 and u = vq2ν1 q3λ1 (see (5.2)). Therefore (1) 1,0 the irreducible quotient N∅,{ν (v) is isomorphic to the Fock module Fu . Similarly 1 },{λ1 } (2)

0,1 the MacMahon module N{µ t (v) is isomorphic to the Fock module Fu , where 1 },∅,{λ1 } µ1 λ1 u = vq1 q3 . n,n The highest weight of the module Nµ,ν,λ (v) given by the rational function which can n,n be decomposed as the product of several factors of the type (5.5). Therefore Nµ,ν,λ (v) is isomorphic to subquotient of Fock module, which can be defined as tensor product of basic ones Fu(i11 ) ⊗ Fu(i22 ) ⊗ · · · ⊗ Fu(ikk ) . ¨ 1 ) in these representations. Now we will describe image of U~q(gl (i)

(i)

5.3. First we consider tensor product Fu1 ⊗ Fu2 . This tensor product was essentially elaborated in paper [12] which we follow. We introduce the Heisenberg generators hn (i) (i) acting on Fu1 ⊗ Fu2 −r/2

h−r = qi−r (a−r ⊗ 1) − qi

(1 ⊗ a−r ),

r/2

hr = qi (ar ⊗ 1) − qir (1 ⊗ ar ),

r > 0.

(5.6)

22


These operators satisfy [hn , ∆(Hm )] = 0 and this condition determines them up to normalization. In our normalization we have r/2

[hr , hs ] = r

−r/2 2

(qir − qi−r )(qi − qi −κr

)

δr+s,0 .

b1 = Q b ⊗ 1, Q b2 = 1 ⊗ Q, b u1 = ep1 , u2 = ep2 . Following [8] we introduce two Denote Q screening currents S+ii (z)

=e

S−ii (z) = e

ǫi+1 b 1 −Q b2) (Q ǫi

ǫi−1 b 1 −Q b2) (Q ǫi

z z

p2 −p1 +ǫi ǫi−1

p2 −p1 +ǫi ǫi+1

exp exp

∞ r/2 −r/2 X −(qi+1 −qi+1 ) r/2

r=1 ∞ X r=1

−r/2

r(qi −qi

)

r/2

−r/2

r/2

−r/2

−(qi−1 −qi−1 ) r(qi −qi

)

h−r z

r

h−r z r

!

!

exp exp

∞ r/2 −r/2 X (qi+1 −qi+1 ) r/2

r=1 ∞ X r=1

−r/2

r(qi −qi r/2

)

hr z

−r

−r/2

(qi−1 −qi−1 ) r/2

−r/2

r(qi −qi

hr z −r ) (5.7)

¨ 1 ) (including the operator d) commutes with the following Lemma 5.1. The image of U~q(gl screening operators I I ii ii ii S+ = S+ (z)dz, S− = S−ii (z)dz. (5.8) This lemma follows from a direct computation. The algebra which commute with the screening operators are usually called [8] the q-deformed W -algebra. The commutativity with screening operators (5.12) determine ˙ 2 . It can be proved that the image W -algebra, which is called q-deformed W -algebra of gl (i) (i) ¨ 1 ) in the representation Fu1 ⊗ Fu2 coincide with the q-deformed W -algebra of gl ˙ 2, of U~q(gl ˙ or W~q(gl2 ) for short. Note that the screening currents formally commute S+ii (z)S−ii (w) =

1 (z −

−1/2 qi w)(z

−

(i)

1/2 qi w)

:S+ii (z)S−ii (w):= S+ii (w)S−ii (z).

(i)

s t For generic u1 , u2 the module Fu1 ⊗ Fu2 is irreducible. But if u2 = u1 qi+1 qi−1 where (i) (i) s, t ∈ Z and st > 0 then it is not so. If s, t > 0, then Fu1 ⊗ Fu2 has irreducible factor 2,0 isomorphic to MacMahon module N∅,{ν (v) (one can see this from comparison 1 ,ν1 },{λ1 ,λ2 } of highest weights). Moreover, this MacMahon module can be written as a cohomology of complex consisting of Fock modules. The simplest example of such complex given if s or t equal to 1

0→F 0→F

(1) ν −1 λ vq2 1 q3 1

(1) ν

λ −1

vq2 1 q3 1

⊗F ⊗F

(1) ν λ −s vq2 1 q3 1

(1) ν −s λ1 q3

vq2 1

11 S+

−−→ F 11 S−

−−→ F

(1) ν

λ

⊗F

λ

⊗F

vq2 1 q3 1 (1) ν

vq2 1 q3 1

(1) ν −1 λ1 −s q3

vq2 1 (1)

ν −s λ1 −1 q3

vq2 1

2,0 → N∅,{ν (v) → 0 1 ,ν1 },{λ1 ,λ1 −s+1} . 2,0 → N∅,{ν1 ,ν1 −s+1},{λ1 ,λ2 } (v) → 0

!

!

.


23

2,0 For generic module N∅,{ν (v) the corresponding short exact sequence have the 1 ,ν1 },{λ1 ,λ2 } form

0→F 0→F

(1) ν −1 λ vq2 2 q3 1

(1) ν

λ −1

vq2 1 q3 2

⊗F ⊗F

11 )ν1 −ν2 +1 (S+

(1) ν λ −1 vq2 1 q3 2

−−−−−−−→ F 11 )λ1 −λ2 +1 (S−

(1) ν −1 λ1 q3

vq2 2

−−−−−−−−→ F

(1) ν

λ

vq2 1 q3 1 (1) ν

λ

vq2 1 q3 1

⊗F ⊗F

(1) ν −1 λ2 −1 q3

vq2 2 (1)

ν −1 λ2 −1 q3

vq2 2

2,0 → N∅,{ν (v) → 0 1 ,ν1 },{λ1 ,λ2 }

→

2,0 N∅,{ν (v) 1 ,ν1 },{λ1 ,λ2 }

,

→0

(5.9) where operators (S±11 )r should be considered as a r-fold integral over the appropriate cycle with the appropriate additional factor. Now we compute the Euler characteristic of (5.9). (1) (1) Using χ(Fu1 ⊗ Fu2 ) = q ∆1 (p1 )+∆1 (p2 ) /(q)2∞ , where ∆i (p) is defined in (5.4) we have q −(λ1 +1)(ν1 +1)−λ2 ν2 − q −(λ1 +1)ν1 −λ1 )(ν2 +1) 2,0 , χ N∅,{ν1,ν1 },{λ1 ,λ2 } (v) = q ∆ (q)2∞

where

∆ = −∆1 (p + ν1 ǫ2 + λ1 ǫ3 ) − ∆1 (p + (ν2 − 1)ǫ2 + (λ2 − 1)ǫ3 ) + (λ1 + 1)(ν1 + 1) + λ2 ν2 , and ep = v. Up to factor q ∆ this formula coincide with (2.6) (or with its special case (1.4)). 0,2 In a similar manner one can construct resolutions of N{µ t (v) in terms of 1 ,µ2 },∅,{λ1 ,λ2 } (2)

(2)

F u1 ⊗ F u2 . Below we will discuss the algebra of screening operators which commute with image of (1) ¨ 1 ) in the representation Fu(1) algebra U~q(gl 1 ⊗ . . . Fun . This system of screening operators coincides with one studied in [8], the algebra which commutes with them (i.e. image of ¨ 1 )) is W~q(gl ˙ n ). We claim that one can construct resolution of N n,o (v) in terms of U~q(gl ∅,ν,λ (1)

(1)

the modules Fu1 ⊗ . . . ⊗ Fun . See Section 5.5 for more details. (1)

(2)

5.4. Second we consider the tensor product Fu1 ⊗ Fu2 . We introduce the Heisenberg (i) (i) generators hn acting on Fu1 ⊗ Fu2 −r/2

h−r = hr =

q1

r/2

−r/2

(q2 − q2

r/2

−r/2

q1 − q1 r/2

−r/2

r/2

−r/2

q2 − q2 q1 − q1

)

(a−r ⊗ 1) − r/2

(ar ⊗ 1) −

r/2

− q1

r/2

− q2

q1 q2

r/2

−r/2

−r/2

−r/2

q2 (q1 − q1 r/2

−r/2

q2 − q2

)

(1 ⊗ a−r ), r > 0.

(5.10)

(1 ⊗ ar ),

These operators satisfy [hn , ∆(Hm )] = 0 and this condition determines them up to normalization. In our normalization we have [hr , hs ] = rδr+s,0. b1 = Q b ⊗ 1, Q b2 = 1 ⊗ Q, b u1 = ep1 , u2 = ep2 and introduce Similar to previous case, denote Q screening current ! ! ∞ ∞ −r/2 r/2 X X ǫ2 b ǫ1 b p2 −p1 +ǫ2 q q Q − Q 2 2 1 2 12 r −r ǫ2 S (z) = e ǫ1 z ǫ3 exp . (5.11) h−r z exp hr z −r r r=1 r=1

24


¨ 1 ) (including the operator d) commutes with the following Lemma 5.2. The image of U~q(gl screening operator I 12 S = S 12 (z)dz. (5.12) This lemma follows from direct computation. Note that in this case we have only one screening current contrary to two currents S−ii (z), S+ii (z) above. Also note that the commutation relations of S 12 (z) do not depend on q1 , q2 , q3 . S 12 (z)S 12 (w) = (z − w) :S 12 (z)S 12 (w):= −S 12 (w)S 12(z), i.e. we have a fermion screening current. In particular we have S 12 S 12 = 0. We will call ˙ 1|1 or the algebra which commute with the operator S 12 by q-deformed W -algebra of gl ˙ 1|1 ) for short. The arguments for such name will be given below. W~q(gl (i) (i) For generic u1 , u2 the module Fu1 ⊗ Fu2 is irreducible. But in the resonance case we have a nontrivial intertwining operators between such modules and can construct complex 1,1 with cohomology N{µ (v). 1 },{ν1 },∅ Namely, for ν1 ≥ µ1 we have exact sequence S 12

(1)

(2)

S 12

(1)

S 12

(2)

(1)

(2)

. . . −−→ Fvq−2 qν1 ⊗Fvq3 qµ1 −−→ Fvq−1 qν1 ⊗Fvq2 qµ1 −−→ Fvqν1 ⊗Fvq 2

3

1 3

2

3

1 3

3

µ1 1 q3

1,1 → N{µ (v) → 0, 1 },{ν1 },∅ (5.13)

and for ν1 ≤ µ1 we have exact sequence S 12

(1)

1,1 0 → N{µ (v) −−→ Fvq 1 },{ν1 },∅

(2)

S 12

(2)

(1)

S 12

(1)

(2)

S 12

⊗Fvqµ1 −−→ Fvq2 qν1 ⊗Fvq−1 qµ1 −−→ Fvq3 qν1 ⊗Fvq−2 qµ1 −−→ . . . 2 3 2 3 3 1 3 1 3 (5.14) (Note that for ν1 = µ1 both complexes exist). Taking the euler characteristics we get 1,1 χ N{µ1 },{ν1 },∅ (v) = q ∆ R(ν1 − µ1 , q) (5.15) ν1 2 q3

where ep = v and ∆ = −∆1 (p + ν1 ǫ3 ) − ∆2 (p + ǫ1 + µ1 ǫ3 ). Up to factor q ∆ this formula coincide with (2.6) for this special case. 5.5. Now we want to consider tensor products ⊗ Fu(2) ⊗ . . . ⊗ Fu(2) . Fu(1) ⊗ . . . Fu(1) n n+1 n+m 1

(5.16)

n,m Similar to previous discussion we expect that the MacMahon module Nµ,ν,λ (v) is given as cohomology of resolutions consisting of modules of the type (5.16). For example one can n/2 m/2 n,m easily see that the central charges of Nµ,ν,λ (v) and (5.16) are equal to c = q1 q2 . Moreover we can consider another ordering in tensor product (5.16). For generic parameters u any tensor product of n modules F (1) and m modules F (2) is isomoprhic to (il+1 ) (i ) the product in order (5.16). For each pair of neighbor Fock modules Full ⊗ Ful+1 we ii

constructed before the screening operators S∗l l+1

l,l+1

, where indices l, l+1 label Fock

modules in which this operator act and ∗ = ± if il 6= il+1 . For any l, ∗ the operator il il+1 ¨ 1 ). commutes with the image U~q(gl S∗ l,l+1


25

For modules (5.16) it is convenient to decompose the corresponding screening operators on 3 systems: n o S1 = S−11 i,i+1 |0 < i < n , n o 22 S2 = S+ j,j+1 |n < j < m + n , (5.17) n o S3 = S+11 i,i+1 , S 12 n,n+1 , S−22 j,j+1 |0 < i < n, n < j < n + m .

˙ n|m ) the algebra which commutes with this system of screening We will denote by W~q(gl operators. Different orderings in (5.16) correspond to different Borel subalgebras in gln|m . Now discuss representation theoretic interpretation of character formulas (2.6), (2.4), (2.3). • Each term in the sum of right side of (2.6) has the form of q ∆ /(q∞ )n+m . This is the character of the Fock module (5.16). Therefore it is natural to expect that right side of (2.6) is an Euler characteristic of a resolution, which consist of Fock modules (5.16). The terms in this resolution should be labeled by (σ, τ, A) ∈ Θ as in (2.6). We will say that this resolution is a materialization of a formula (2.6). This resolution is a generalization of resolutions (5.9) and (5.13),(5.14) discissed above. The intertwining operators in this resolution could be constructed using screening operators. The construction of such resolution is unknown (actually we didn’t give proof of the existence of (5.9) and (5.13),(5.14) but these particular cases are rather easy). ˙ n ) we didn’t find this resolution in Even in the case m = 0 which correspond to W~q(gl the literature. It is well known that conformal limit of such resolution exists. For the ˙ n ) case see [11]. construction of intertwing operators in the W~q(gl ˙ n|m ). The algebra which commute • There exist another class of representations of W~q(gl n o with screening operators S±11 i,i+1 , S±22 j,j+1 |0 < i < n, n < j < n + m (the same set ˙ n ) ⊗ W~q (gl ˙ m ). Therefore as before except (S 12 )n,n+1) is isomorphic to the product W~q (gl ˙ n|m ) is a subalgebra of W~q(gl ˙ n ) ⊗ W~q(gl ˙ m ). The last algebra has represenalgebra W~q (gl n,0 0,m tations of the form N∅,ν,ν ′ (v1 ) ⊗ Nµ,∅,µ′ (v2 ). Since the character of each factor is given by formula (1.4) we have n,0 χ(N∅,ν,ν ′ (v1 )

⊗

0,m Nµ,∅,µ ′ (v2 ))

=q

∆ aν+ρn (q

−ν ′ −ρn

′

)aµ+ρn (q −µ −ρn ) , n+m (q)∞

for certain ∆. The right side of character formula (2.4) is a linear combination of such terms. Therefore it is natural to expect that there exists a resolution consisting of modules n,0 0,m n,m N∅,ν,ν This resolutions should be a ′ (v1 ) ⊗ Nµ,∅,µ′ (v2 ) with the cohomology Nµ,ν,λ (v). materialization of character formula (2.4). See also Section 5.7 below. • Consider tensor product (1) (1) (2) (2) (1) (2) (1) (2) Fu1 ⊗. . . Fun−r ⊗ Fun−r+1 ⊗. . .⊗Fun+m−2r ⊗ Fun+m−2r ⊗Fun+m−2r+2 ⊗. . . Fun+m−1 ⊗Fun+m . (5.18) As we mention before this product is isomorphic to (5.16).

26


Consider the following system of screening operators n o S 12 l,l+1 |n + m − 2r < l < n + m, l ≡ n + m + 1 mod 2 .

˙ 1 )⊗(n+m−2r) ⊗ W~q(gl ˙ 1|1 )⊗r , The W -algebra, which commutes with this system is W~q (gl ˙ 1 )⊗n+m−2r is just Heisenberg algebra acting on first n + m − 2r factors on where W~q(gl (5.18). But this system is a subsystem of all screening operators acting (5.18), therefore ˙ n|m ) is a subalgebra of W~q (gl ˙ 1 )⊗n+m−2r ⊗ W~q(gl ˙ 1|1 )⊗r . The last W -algebra has the W~q(gl representations in the space 1,1 1,1 Fu(1) ⊗. . . Fu(1) ⊗ Fu(2) ⊗. . .⊗Fu(2) ⊗N{m (v1 )⊗. . .⊗N{m (vr ). (5.19) 1 n−r n−r+1 n+m−2r r },{nr },∅ 1 },{n1 },∅ Q Due to (5.15) the character of this representations equals ri=1 R(di ; q) · q ∆ /(q∞ )m+n−2r , where di = ni − mi and certain ∆. in right side of (2.3) we get the linear combination of terms QrIf we compute∆determinant m+n−2r . Therefore it is natural to conjecture the existence of the i=1 R(di ; q) · q /(q∞ ) n,m resolution of Nµ,ν,λ (v) which consists of modules of the type (5.19). And this resolution should be a materialization of the character formula (2.3).

5.6. Now we consider conformal limit of the previous construction. We rescale ǫi → ~ǫi , pl → ~pl and then send ~ to zero i.e. send all parameters qi , ul to 1 with the certain speed. The limit of screening operators introduced above is well defined. Denote by a ¯n,l the limit of generators an acting on the l-th factor in (5.16). As the limit of (5.3) we get [¯ar,i , a ¯s,i ] = −r

ǫ2 ǫ21 δr+s,0 , [¯ar,j , a ¯s,j ] = −r 2 δr+s,0 , 1 ≤ i ≤ n, n + 1 ≤ j ≤ n + m, ǫ2 ǫ3 ǫ1 ǫ3

¯n,l forms the Heisenberg algebra and [¯ar,l , a ¯s,l′ ] = 0, for l 6= l′ . In other words operators a n+m constructed from space C with the scalar product (·, ·) given in orthogonal basis el by formula: (ei , ei ) = −

ǫ21 ǫ2 , (ej , ej ) = − 2 , 1 ≤ i ≤ n, n + 1 ≤ j ≤ n + m. ǫ2 ǫ3 ǫ1 ǫ3

It is convenient to introduce ϕl (z) =

X a ¯r,l −r ¯l, z +a ¯0,l log z + Q r

r∈Z\0

¯ l is a limit Q bl , with relation [a0,l , Q ¯ l ] = (el , el ). where Q The limit of screening currents have the form Xn+m 11 lim(S± (z))i,i+1 = exp (α±,i )l ϕl (z) , 1 ≤ i ≤ n, l=1 ~→0 Xn+m lim (S 12 (z))n,n+1 = exp (αn )l ϕl (z) l=1 ~→0 Xn+m (α±,j )l ϕl (z) , n + 1 ≤ j ≤ n + m. lim (S±22 (z))j,j+1 = exp ~→0

l=1

(5.20)


27

We consider the vectors α±,i , αn , α±.j as the vectors in Cn+m and they have the form ǫ2 ǫ2 ǫ1 ǫ1 ǫ1 ǫ2 α−,j = ej − ej+1 α+,i = ei − ei+1 , αn = en − en+1 , ǫ1 ǫ1 ǫ1 ǫ2 ǫ2 ǫ2 ǫ3 ǫ3 ǫ3 ǫ3 α−,i = ei − ei+1 , α+,j = ej − ej+1 . ǫ1 ǫ1 ǫ2 ǫ2 Slightly abusing notation we will say that α ∈ SI , for I = 1, 2, 3 if the corresponding screening operator belongs the SI . For any β, γ ∈ Cn+m the commutation relations of vertex operators have the form Xn+m Xn+m Xn+m βl ϕl (z) + γϕl (w) γl ϕl (w) = (z − w)(β,γ) exp βl ϕl (z) exp exp l=1

l=1

l=1

In particular, if (β, γ) ∈ 2Z then the corresponding vertex operators formally commute and if (β, γ) ∈ 2Z + 1 then the corresponding vertex operators formally anticommute. It is easy to see that if two vectors α, α′ belong to different systems S then the scalar product (α, α′) ∈ Z. And the Gramian matrices for vectors from S are given below  −2ǫ3  −2ǫ3   ǫ3 ǫ3 0 ... 0 0 ... 0 ǫ2 ǫ2 ǫ1 ǫ1 ..  ..   ǫ3  ǫ3 .. .. .. .. −2ǫ3 −2ǫ3 . . . . .  .   ǫ2  ǫ1 ǫ ǫ 2 1     .. .. .. .. .. ..  0  0   . . . . . . 0 0  , S2 :  , S1 :   .    .. .. . . −2ǫ3 .. . . −2ǫ3 ǫ ǫ  ..    3 3 . . . . .   ǫ2 ǫ2  ǫ1 ǫ1      −2ǫ3 −2ǫ3 ǫ3 ǫ3 0 ... 0 0 . . . 0 ǫ2 ǫ2 ǫ1 ǫ1  −2ǫ

          S3 :          

ǫ3 ǫ2 ǫ3

0 .. . 0 .. . .. . .. . 0

2

ǫ2 ǫ3 −2ǫ2 ǫ3

..

.

..

. ...

0 .. . .. . .. . 0

... .. . .. .

0 .. .

−2ǫ2 ǫ3 ǫ2 ǫ3

ǫ2 ǫ3

..

...

...

.

...

...

...

...

... .. . .. .

0 ..

.

ǫ1 ǫ3

ǫ1 ǫ3 −2ǫ1 ǫ3

0 .. .

..

.

..

.

0 .. . .. . .. .

...

0

1

0

−2ǫ1 ǫ3 ǫ1 ǫ3

0 .. . .. . .. .



         0  . ..  .    0    ǫ1  ǫ3   −2ǫ1 ǫ3

The Gramian matrix corresponding to S1 is equal to the Cartan matrix of sln multiplied by − ǫǫ32 . The W -algebra commuting with screening operators with such Gramian ˙ n ) ⊗ Heis⊗m since there matrix is called W(sl˙ n ) [6]. In our case the W algebra is W(gl exists m + 1 dimensional Heiseberg algebra which commutes with all screening operators from S1 . Similarly, commutativity with screening operators from S2 determines W ˙ m ). algebra Heis⊗n ⊗ W(gl The Gramian matrix corresponding to S3 have blocks corresponding to sln , slm and fermionic screening operator between them. Therefore we call the W -algebra commuting

28


˙ n|m ). The W(gl ˙ n|1 ) case was considered in [15], but we didn’t find with this system W(gl any reference for general n, m. Note that our W -algebras differ from ones introduced in [17].

5.7. Standard statement in the theory of vertex algebras is an equivalence of the abelian categories of certain representations of vertex algebra and certain representations of quantum group. This is a statement similar to Drinfeld–Kohno or Kazhdan–Lusztig theorem. In fact this equivalence of categories is an equivalence of braided tensor categories but we do not need tensor structure here. ˙ n ) is equivIt is known that certain category of representations of vertex algebra W(gl alent to certain category of representations of quantum group Uq gln ⊗ Uq′ gln , where parameters q, q ′ are given in terms of ǫ1 , ǫ2 (people also use modular double of Uq gln ). We ˙ n|m ) and quantum group conjecture that the same relation holds for vertex algebra W(gl Uq gln|m ⊗ Uq′ gln ⊗ Uq′′ glm for certain q, q ′, q ′′ given in terms of ǫ1 , ǫ2 . (n) Denote by Lν the finite dimensional irreducible representation of Uq′ (gln ), recall that these representations are labeled by partitions ν such that l(ν) ≤ n. Similarly denote (m) by Lµ the finite dimensional irreducible representation of Uq′′ gln . For Uq gln|m consider only tensor irreducible representations i.e. irreducible submodules of the tensor powers of Cn|m . They are labeled by partitions λ such that λn+1 < m + 1 [2],[25] and we denote (n|m) them by Lλ . There are also Kac modules for Uq gln|m which are induced from the (n) (m) representation of parabolic subalgebra p on the tensor product representation of Lν ′ ⊗Lµ′ Uq gln ⊗ Uq glm ⊂ p ⊂ Uq gln|m . We denote such modules by Vν ′ ,µ′ . We conjecture that under equivalence above the tensor product of irreducible mod(n|m) (n) (m) n,m ules Lλ ⊗ Lν ⊗ Lµ goes to the conformal limit of Nµ,ν,λ (v). And tensor product (n) (m) n,0 0,m Vν ′ ,µ′ ⊗ Lν ⊗ Lµ goes to the conformal limit of N∅,ν,ν ′ (v1 ) ⊗ Nµ,∅,µ ′ (v2 ). In paper [5] Cheng, Kwon and Lam constructed a resolution in terms of Kac modules of the tensor module of gln|m . Taking the conjectural q-deformation of this resolution we (n|m) have a complex which consist of modules Vν ′ ,µ′ with the cohomology Lλ . Multiplying (n) (m) by Lν ⊗ Lµ and applying equivalence we get the resolution of the conformal limit of 0,m n,m n,0 Nµ,ν,λ (v) in terms of conformal limit of N∅,ν,ν ′ (v1 ) ⊗ Nµ,∅,µ′ (v2 ). This resolution should be a materialization of (2.4), its q-deformation was discussed above in Section 5.5. The Euler characteristic of resolution constructed in [5] yields the following formula

sλ (x|y) =

X

(−1)

α1 ≥α2 ≥...≥αr ≥r−m

P

αi

yj 1+ sπ+m−r,−α (x)sα,κ (y) . xi 1≤i≤n,1≤j≤m Y

Here notations π, κ were introduced in Section 2.1, sµ (x) is a Schur polynomial i.e. (m) character of Lµ and sλ (x|y) is a hook Schur polynomial (or super-Schur polynomial) i.e. (n|m) character of Lλ . This formula resembles our character formula (2.4).


29 (n|m)

Remark 5.1. Moens and van der Jeugt found another formula for the character of Lλ   Q yj P mn−r Qj a −a−1+m a (−1) 1+ (−1) xj yi yi  1≤i≤m  xi i,j 1≤i≤m a≥0 1≤j≤m−r  . det  sλ (x|y) = 1≤j≤n   V (x1 , . . . , xn )V (y1 , . . . , ym ) xPj i +m 1≤i≤n−r 0 1≤j≤n

(5.21) This formula is similar to our formula (2.3). It is natural to conjecture that there is a resolution which is a materialization of (5.21) and under the equivalence this resolution goes to resolution which is materialization of (2.3).

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