Bosonic and Fermionic Representations of Endomorphisms of Exterior Algebras
aa r X i v : . [ m a t h . R T ] S e p Bosonic and Fermionic Representations of Endomorphisms ofExterior Algebras
Ommolbanin Behzad, Letterio Gatto ∗ Abstract
We describe the fermionic and bosonic Fock representation of the Lie super-algebra of endomor-phisms of the exterior algebra of the Q -vector space of infinite countable dimension, vanishingat all but finitely many basis elements. We achieve the goal by exploiting the extension of theSchubert derivations on the fermionic Fock space. Introduction
Let B := Q [ x ] be the polynomial ring in the infinitely many indeterminates x :=( x , x , . . . ). The purpose of this paper is to further enhance a classical but fundamental result byDate, Jimbo, Kashiwara and Miwa, which describes the polynomial algebra B ( ξ ) := B ⊗ Q Q [ ξ − , ξ ](the bosonic Fock space ) as a representation of the Lie algebra gl ∞ ( Q ) := { ( a ij ) i,j ∈ Z | all but a finite number of the a ij ∈ Q are zero } .For the convenience of our exposition, we slightly change the notation for gl ∞ ( Q ) as follows. Let usconsider the Q -vector space V := L i ∈ Z Q · b i , whose basis b = ( b i ) i ∈ Z is parametrised by the integers,along with its restricted dual V ∗ := L i ∈ Z Q · β j , with basis β = ( β i ) i ∈ Z , where β j ∈ Hom Q ( V , Q ) isthe unique linear form such that β j ( b i ) = δ ji . In this way, the algebra gl ∞ ( Q ) gets identified with gl ( V ) = V ⊗ V ∗ = M i,j ∈ Z Q · b i ⊗ β j . (1)In the contribution [2] (see also [9]) Date, Jimbo, Kashiwara and Miwa compute the action on B ( ξ )of the generating function E ( z, w ) = X i,j b i ⊗ β j z i w − j (2)of the basis as in (1), and obtain their celebrated bosonic vertex operator representation of gl ( V ).Our main result is the ultimate extension of the DJKM formula, concerned with the fermionic andthe bosonic vertex representation of the Lie super–algebra gl ( V V ) := V V ⊗ V V ∗ , where V V and V V ∗ denote, as usual, the exterior algebra of V and V ∗ respectively. Our formula includes, andadmits as a particular case, the DJKM one, as displayed, e.g., in [10, Proposition 5.2] or [9, Section1]. ∗ Work sponsored by Finanziamento Diffuso della Ricerca, no. 53 RBA17GATLET del Politecnico di Torino;Progetto di Eccellenza Dipartimento di Scienze Matematiche, 2018–2022 no. E11G18000350001, INDAM-GNSAGAe PRIN ”Geometria delle Variet`a Algebriche”.
Keywords and Phrases:
Schubert Derivations on the fermionic Fock space, Vertex Operators on Exterior Algebras,Bosonic and Fermionic Representations by Date-Jimbo-Kashiwara-Miwa, Symmetric Functions. .2 Let P be the set of all partitions (non increasing sequences of non negative integers all zerobut finitely many). The fermionic Fock space is a Z -graded vector space F := L m ∈ Z F m which,like B ( ξ ), the bosonic one, possesses a basis [ b ] m + λ parametrised by Z × P . More than that,it is essential, from our point of view, to think of F as a B ( ξ )-module of rank 1 generated by[ b ] := b ∧ b − ∧ b − ∧ · · · , such that ξ m S λ ( x )[ b ] = [ b ] m + λ = b m + λ ∧ · · · ∧ b m − r +1+ λ r ∧ b m − r ∧ b m − r − ∧ · · · , (3)where S λ ( x ) denotes the Schur polynomial associated to the partition λ and to the sequence x .Equality (3) can be understood either as a Giambelli’s formula for Schubert Calculus on infiniteGrassmannian (see [7]) or like a Jacobi-Trudy formula. To follow more closely the reference [10,Theorem 6.1], and being more adherent to the subject of the paper, we call (3) the Boson-Fermioncorrespondence . Our starting point is the obvious remark that V V is a (irreducible) representa-tion of the Lie super–algebra gl ( V V ) of all endomorphisms vanishing at all basis elements butfinitely many of the exterior algebra. An explicit generating function encoding the gl ( V V )-modulestructure of V V has already been proposed in [1], where the vertex operators shaping the boson-fermion correspondence spontaneously arise in all their splendor, regardless of the more classicalframework. In addition, as noticed in [8], little effort is needed to extend the V V -representationto F , mainly because the latter is a module over the former. This reflects in the fact that eachdegree F m of F , as suggested in formula (3), can be thought of as a semi-infinite exterior power.Finally, one just pulls back on B ( ξ ) the F representation of gl ( V V ), invoking the boson-fermioncorrespondence. The program demands, however, to identify a basis of V V ⊗ V V ∗ suited to get aconvenient generalisation of the DJKM generating function (6). Last, but not the least, one is leftto determine explicitly its action on V V . This is the point that, as in our previous contribution,the flexible formalism of Schubert derivations (a distinguished kind of Hasse-Schmidt derivation onan exterior algebra), extended to F , enters the game. To pursue our program we use the basis of V V ⊗ V V ∗ = L k,l ≥ V k V ⊗ V l V ∗ obtained as theunion of those induced on V k V ⊗ V l V ∗ by b and β , for all k, l ≥
0. This is quite straightforward,up to getting aware of one main combinatorial point, i.e. that they are best parametrised by theset P of what, in Definition 2.1, lacking of a better terminology, we called bilateral partitions . Moreprecisely, given r ≥
0, we shall understand by P r the set of all r -tuples λ = ( λ , . . . , λ r ) ⊆ Z r , suchthat λ ≥ · · · ≥ λ r . We so have k ^ V = M µ ∈P k Q [ b ] k µ and l ^ V ∗ = M ν ∈P l Q [ β ] l µ , where [ b ] k µ = b k − µ ∧ · · · ∧ b µ k and [ β ] l ν = β l − ν ∧ · · · ∧ β ν l . Then E ( z k , w − l ) = X µ , ν ∈P k ⊗P l [ b ] k µ ⊗ [ β ] l ν s µ ( z k ) s ν ( w − l ) , is the generating function of the distinguished basis [ b ] k µ ⊗ [ β ] l ν of V k V ⊗ V l V ∗ , where z k and w − l are, respectively, k -tuples ( z , . . . , z k ) and l -tuples ( w − , . . . , w − l ) of formal variables. Abusingnotation, we have chosen to denote by the same symbols s µ ( z k ) and s ν ( w − l ) natural extensions ofthe classical Schur polynomials occurring in the theory of symmetric functions as in, e.g., [4, Section3] and/or [3, Section 2.2.]. The difference with the classical ones is that they are symmetric rational unctions and do coincide with the usual Schur symmetric polynomials whenever λ ∈ P r = P ∩ N r .We are now in position to anticipate the statement of our main result. Theorem 3.6.
The (DJKM bosonic) action of E ( z k , w − l ) on B ( ξ ) is given by E ( z k , w l ) = exp X n ≥ n p n ( z − k ) p n ( w l ) Γ( z k , w l ) , (4) wherei) the map Γ( z k , w l ) : B ( ξ ) → B ( ξ ) J z ± k , w ± l K is the vertex operator R ( z k , w − l )exp X n ≥ x n ( p n ( z k ) − p n ( w l )) exp X n ≥ p n ( z − k ) − p n ( w − l ) n ∂dx n ; (5) ii) the map R ( z k , w − l ) : B ( ξ ) J z k , w − l K → B ( ξ ) J z k , w − l K is the unique B J z k , w − l K -linear exten-sion of ξ m ξ m + k − l Y ≤ i ≤ k ≤ j ≤ l z m − l +1 i w m − l +1 j ; iii) the expression p n ( z ± k ) and p n ( w ± l ) denote the Newton powers sums symmetric polynomials,in the variables z ± k and w ± l , i.e. more explicitly p n ( z ± k ) := z ± n + · · · + z ± nk and p n ( w ± l ) := w ± n + · · · + w ± nl . The meaning of formula (5) is that if P ( x , ξ ) ∈ B ( ξ ) is any polynomial, then the “multiplica-tion” of [ b ] k µ ⊗ [ b ] l ν is the coefficient of s µ ( z k ) s ν ( w − l ) in the expansion E ( z k , w − l ) P ( x , ξ ). Thismay seem tricky. However multiplying the resulting expression by the product of the Vander-monde ∆ ( x k )∆ ( w − l ), it is sufficient to consider the coefficient of the less intimidating monomial z k − µ k · · · z µ k · w − l +1 − ν · · · w − ν k l suffices.To end up, reading formula (5) for k = l = 1, putting z = z and w = w , one has s ( i ) ( z ) = z i and s ( j ) ( w − ) = w − j , for all i, j ∈ Z . By the definition of the logarithm of an invertible formalpower series: exp X n ≥ n w n z n = 11 − wz and the fact that, in this case, R ( z, w − ) ξ m = ξ m z m w m , equality (4) simplifies into E ( z, w ) | Bξ m = z m w m − wz exp X n ≥ x n ( z n − w n ) exp − X n ≥ z − n − w − n n ∂∂x n , (6)which is precisely the original DJKM formula for the bosonic representation of gl ( V ) (see e.g. [10,Proposition 5.2] or [9, Section 1]. This may look surprising indeed, because comparing (5) with(6), it is apparent that (5) can be obtained from the DJKM expression simply by replacing thevariables z, w in (6) by the power sums of the k and l -tuples of indeterminates needed to write3he appropriate generating functions. As in our previous references [1, 7, 8], we have borrowedmethods from the theory of Hasse-Schmidt derivation on a exterior algebra, like in the book [5].The similarity of DJKM formula with our (4), however, makes us wonder whether there is anyother argument to deduce our Theorem 3.6 bypassing our methods. In the first section we recall some more or less known pre-requisites. We revise, in particular, the construction of the fermionic Fock space following [8,Section 5] as well as how to extend the Schubert derivation on it. A little background on Schurpolynomials, mainly following [4] but also [10, Lecture 6], is included as well. Section 2 is devotedto carefully define the generating function of the basis elements of V k V ⊗ V l V ∗ , that is bestsuited to describe the fermionic and bosonic representation of gl ( V V ). In this same section thenatural notion of bilateral partition is also introduced. It is reasonable to suspect it somewherehidden in some less known literature. Section 3 eventually concerns the statement and proof of ourmain theorem which, as announced, supplies the expression of both the fermionic and the bosonicexpression of gl ( V V ). The two cases are treated in a unified way, reflecting the fact inspiring thereferences [5, 7, 8] that there is a very little, if not any at all, substantial difference between the twospaces. Indeed, as explained in [1], the vertex operators occurring in the representation theory ofthe Heisenberg algebra, come naturally to life, exactly the same, already at the level of multivariateSchubert derivations on exterior algebras. With no serious need, at least for the focused purposesof our research, to cross the walls to enter in the realm of the infinite wedge powers, as however wedid in the present contribution. We shall deal with a Q -vector space V := L i ∈ Z Q · b i and its restricted dual V ∗ := L i ∈ Z Q · β j ,where β j ∈ Hom Q ( V , Q ) is the unique linear form such that β j ( b i ) = δ ji . The generating series ofthe basis elements of V and V ∗ are, respectively: b ( z ) = X i ∈ Z b i z i ∈ V J z − , z K and β ( w − ) = X j ∈ Z β j w − j ∈ V ∗ J w, w − K . (7) V V . A map D ( z ) : V V → V V J z K is said to be Hasse-Schmidt (HS) derivation on V V if D ( z )( u ∧ v ) = D ( z ) u ∧ D ( z ) v , for all u , v ∈ V V . Write D ( z ) inthe form P j ≥ D j z j , with D j ∈ End Q ( V V ). Then D ( z ) is invertible in End Q ( V V ) J z K if and onlyif D is invertible In this case D ( z ) is invertible and its inverse D ( z ) is a HS–derivation as well. Consider the shifts endomorphisms σ ± ∈ gl ( V V ) given by σ ± b j = b j ± . By [5, Proposition 4.1.13], there exist unique HS derivations on σ ± ( z ) : V V → V V J z ± K such that σ ± ( z ) b j = X i ≥ b j ± i z ± i . Let us denote by σ ± ( z ) their inverses in V V J z ± K . Restricted to V they work as follows σ + ( z ) b j = b j − b j +1 z and σ − ( z ) b j = b j − b j − z − . (8)They are called Schubert derivations in the references [5, 7, 8].
We quickly summarise the definition of the fermionic Fock spaceborrowed from [8]. Let [ V ] be a copy of V (framed by square bracket to distinguish by the original V itself). It is the Q -vector space with basis ([ b ] m ) m ∈ Z Identify [ V ] with a sub-module of the tensor4roduct V V ⊗ Q [ V ] via the map [ b ] m ⊗ [ b ] m . Let W be the V V –submodule of V V ⊗ Q [ V ]generated by all the expressions { b m ⊗ [ b ] m − − [ b ] m , b m ⊗ [ b ] m } m ∈ Z . In formulas: W := ^ V ⊗ (cid:0) b m ⊗ [ b ] m − − [ b ] m (cid:1) + ^ V ⊗ (cid:0) b m ⊗ [ b ] m (cid:1) . The fermionic Fock space is the V V - module F := F ( V ) := V V ⊗ Q [ V ] W . (9)Let V V ⊗ Q [ V ] → F be the canonical projection. The class of u ⊗ [ b ] m in F will be denoted u ∧ [ b ] m .Thus the equalities b m ∧ [ b ] m = 0 and b m ∧ [ b ] m − = [ b ] m hold in F . For all m ∈ Z and λ ∈ P let,by definition[ b ] m + λ := b rm + λ ∧ [ b ] m − r = b m + λ ∧ b m − λ ∧ · · · ∧ b m − r +1+ λ r ∧ [ b ] m − r where r is any positive integer such that ℓ ( λ ) ≤ r , which implicitly defines b rm + λ as an element of V r V ≥ m − r +1 , where by V ≥ j we understand L i ≥ j Q · b i . It turns out that F is a graded V V -module: F := M m ∈ Z F m , where F m := M λ ∈P Q [ b ] m + λ = M r ≥ M λ ∈P r Q b rm + λ ∧ [ b ] m − r . (10)is the fermionic Fock space of charge m [10, p. 36]. i) The equality b j ∧ [ b ] m = 0 holds for all j ≤ m ;ii) The image of the map V r V ⊗ F m → F given by ( u , v ) u ∧ v is contained in F m + r . Proof.
They are [8, Proposition 4.4 and 4.5]. F . We now extend the Schubert derivations, inprinciple only defined on V V , on F according to [8] to which we refer to for more details. First wedefine their action on elements of the form [ b ] m by setting: σ − ( z )[ b ] m = σ − ( z )[ b ] m := [ b ] m , σ + ( z )[ b ] m := σ + ( z ) b m ∧ [ b ] m − and σ + ( z )[ b ] m := X j ≥ [ b ] m +(1 j ) z j where (1 j ) denotes the partition with j parts equal to 1. Finally, we set σ ± ( z )[ b ] m + λ = σ ± ( z ) b rm + λ ∧ σ ± ( z )[ b ] m − r and σ ± ( z )[ b ] m + λ = σ ± ( z ) b rm + λ ∧ [ b ] m − r . (11)5 .8 Proposition. For all m ∈ Z , Giambelli’s formula for the Schubert derivation σ + ( z ) holds: [ b ] m + λ = det( σ λ j − j + i )[ b ] m (12) Proof.
See [8, Proposition 5.13].We introduce now an operator on F which, in a sense, plays the role of the determinant ofthe shift endomorphism σ . We denote it by ξ . We shall understand it as the unique algebraendomorphism of V V such that ξ · b j = b j +1 . Being an algebra homomorphism implies that ξ b m + λ = b m +1+ λ It is clearly invertible. Its inverse ξ − is such that ξ − b j = b j − . Secondly, we extend it to F as follows: ξ [ b ] m + λ = ξ ( b rm + λ ) ∧ [ b ] m +1+ λ , (13)where r is any integer greater than the length of the partition λ . It is trivial to check that such adefinition does not depend on the choice of r > ℓ ( λ ). So for instance ξ m ′ [ b ] m + λ = [ b ] m + m ′ + λ . Let B := Q [ x ], the polynomial ring in infinitely many indeterminates x := ( x , x , . . . ). As a Q –vector space it possesses a basis of Schur polynomials parametrised bythe set P of all partitions. Moreover, ( S ( x ) , S ( x ) , . . . ) generate B as a Q -algebra, because S i ( x )is a polynomial of degree i , for all i ≥
0. If λ ∈ P one sets S λ ( x ) = det( S λ j − j + i ( x )) (14)where the sequence ( S ( x ) , S ( x ) , . . . ) is defined by X j ∈ Z S j ( x ) z j = exp( X i ≥ x i z i ) . (15)Let B ( ξ ) := B ⊗ Q Q [ ξ − , ξ ] be the Q [ ξ ]-algebra of B -valued Laurent polynomials in ξ . We shallrefer to B ( ξ ) as the bosonic Fock space. It follows that B ( ξ ) = M m ∈ Z , λ ∈ P Q · ξ m S λ ( x ) The space F can be endowed with a structure of free B ( ξ )-module generated by [ b ] of rankone generated by [ b ] such that ξ m S λ ( x )[ b ] = [ b ] λ , by simply declaring ξ m S i ( x )[ b ] λ := σ i [ b ] m + λ . (16)In fact[ b ] m + λ = ξ m [ b ] λ (Equation (13))= ξ m det( σ λ j − j + i )[ b ] (Giambelli’s formula for Schubert derivations)= ξ m det( S λ j − j + i )[ b ] (by equality (16))= ξ m S λ ( x )[ b ] (Definition of S λ ( x )).6quality (16) can be also phrased by saying that S i ( x ) is an eigenvalue of the Q ( ξ )-linear map σ i : F → F with F m as eigenspaces. It implies that σ + ( z )[ b ] m + λ = exp X i ≥ x i z i [ b ] m + λ , (17)i.e., abusing terminology, exp( P i ≥ x i z i ) is an eigenvalue of σ + ( z ). i) The Schubert derivations σ ± ( z ) , σ ± ( z ) commute with multiplication by ξ , i.e. ξσ ± ( z ) = σ ± ( z ) ξ and ξσ ± ( z ) = σ ± ( z ) ξ ; (18) ii) by regarding the Schubert derivation σ − ( z ) (resp. σ − ( z ) ) as a map B → B [ z − ] by setting ( σ − ( z ) S λ ( x ))[ b ] m = σ − ( z )[ b ] m + λ (resp. ( σ − ( z ) S λ ( x ))[ b ] m = σ − ( z )[ b ] m + λ , one has: σ − ( z ) S i ( x ) = S i ( x ) − S i − ( x ) z (19) σ − ( z ) S i ( x ) = i X j =0 S i − j ( x ) z j ; (20) iii) the maps σ − ( z ) and σ − ( z ) are Q ( ξ ) -algebra endomorphism of B ( ξ ) . In particular σ − ( z ) S λ ( x ) = det( σ − ( z ) S λ j − j + i ( x )) (21) and σ − ( z ) S λ ( x ) = det( σ − ( z ) S λ j − j + i ( x )); (22) iv) the maps σ − ( z ) and σ − ( z ) act on B as exponential of a first order differential operators,namely: σ − ( z ) S λ ( x ) = exp X n ≥ nz n ∂∂x n S λ ( x ) (23) and σ − ( z ) S λ ( x ) = exp − X n ≥ nz n ∂∂x n S λ ( x ) . (24) Proof. i) First we show that the commutation holds on the exterior algebra V V . This is nearlyobvious, because σ ± ( z ) ξb j = σ ± ( z ) b j +1 = X i ≥ b j +1 ± i z ± i = ξ X i ≥ b j ± i z ± i = ξσ ± ( z ) b j The same holds for σ ± ( z ). We have σ ± ( z ) ξb j = σ ± ( z ) b j +1 = b j +1 − b j +1 ± z ± = ξ ( b j − b j ± z ± ) = ξ σ ± ( z ) b j . Secondly, the commutation rules hold for elements of the form [ b ] m . In fact:7 − ( z ) ξ [ b ] m = σ − ( z )[ b ] m +1 (Definition of ξ )= [ b ] m +1 ( σ − ( z ) acts as the identity)= ξ [ b ] m = ξσ − ( z )[ b ] m (Definition of ξ and σ − ( z ) actsas the identity on [ b ] m )Similarly one sees that σ − ( z ) ξ = ξσ − ( z ). The check for σ + ( z ) and σ + ( z ) works analogously asfollows. σ + ( z ) ξ [ b ] m = σ + ( z )[ b ] m +1 (Definition of ξ )= σ + ( z ) b m +1 ∧ [ b ] m (Definition of σ + ( z )[ b ] m )= P i ≥ b m +1+ i z i ∧ [ b ] m (Definition of σ + ( z ) b m )= P i ≥ ξb m + i ∧ ξ [ b ] m − = ξσ + ( z )[ b ] m and σ + ( z ) ξ [ b ] m = σ + ( z )[ b ] m +1 (Definition of ξ )= P j ≥ ( − j b m +1+(1 j ) ∧ [ b ] m − j z j (Definition of σ + ( z )[ b ] m +1 )= P j ≥ ( − j ξb m +(1 j ) ∧ ξ [ b ] m − − j z j (Definition of multiplying by ξ )= ξ P j ≥ ( − j b m +(1 j ) ∧ [ b ] m − − j z j = ξσ + ( z )[ b ] m Let us show now that (18) holds when evaluated against a general element of F . We check for σ + ( z ), the others being analogous and even easier. Let λ be any partition and r any integer suchthat ℓ ( λ ) < r . Then: σ ± ( z )( ξ [ b ] m + λ ) = σ ± ( z )[ b ] m +1+ λ (definition of multiplication by ξ )= σ ± ( z )( b rm +1+ λ ∧ [ b ] m +1 − r ) (decomposition of [ b ] m +1+ λ )= σ ± ( z ) b rm +1+ λ ∧ σ ± ( z )[ b ] m +1 − r ( σ ± ( z ) is a derivation)= σ ± ( z ) ξ b rm + λ ∧ σ ± ( z ) ξ [ b ] m − r (definition of multiplication by ξ )= ξσ ± ( z ) b rm + λ ∧ ξσ ± ( z )[ b ] m − r (Lemma 1.11, item i))= ξσ ± ( z )[ b ] m + λ .The proof for the Schubert derivations σ − ( z ) and σ ± ( z ) works the same.ii) The proof of this second statement works verbatim as in [6, Proposition 5.3], where the S i ( x )are denoted by h i ;iii) In this case the check follows by combining [6, Proposition 7.1] and [6, Corollary 7.3];8v) Recall that B ( ξ ) = Q ( ξ )[ S ( x ) , S ( x ) , . . . ]. Equation (15) implies that ∂S i ( x ) ∂x j = S i − j ( x ) , Then (19), e.g., says that σ − ( z ) S i ( x ) = (cid:18) − z ∂∂x (cid:19) S i ( x ) = exp − X n ≥ nz n ∂ n ∂x n S i ( x ) (25)Now ∂ n ∂x n S i ( x ) = ∂∂x n S i ( x ). Since S i ( x ) generate B as a Q -algebra and σ − ( z ) are algebra homo-morphisms coinciding on generators, (24) follows. The proof of (23) is analogous, but it also followsfrom inverting both members of the equality (24), obtaining σ − ( z ) = exp X n ≥ nz n ∂∂x n In the sequel we will need the following observation. Suppose that φ is any of the endomor-phism σ ± i of σ ± j , for i and j arbitrary non negative integers and that φ [ b ] m + λ = X µ a µ [ b ] m + µ . Then, for any m ′ ∈ Z , X µ a µ [ b ] m + m ′ + µ = φ [ b ] m + m ′ + λ . The proof is based on the definition of multiplication by ξ . X µ a µ [ b ] m + m ′ + µ = X µ a µ ξ m ′ [ b ] m + µ = ξ m ′ X µ a µ [ b ] m + µ = ξ m ′ φ [ b ] m + λ = φξ m ′ [ b ] m + λ = φ [ b ] m + m ′ + λ . V k V and V l V ∗ Let V V = L k ≥ V k V and V V ∗ = L l ≥ V l V ∗ be the exterior algebra of V and V ∗ respectively.To describe the bases of V k V and V l V ∗ induced by the basis b of V and of β of V ∗ (Cf. Section1.1), we need to explain what we shall mean by bilateral partition . A bilateral partition of length at most r ≥ is an element of the set: P r := { λ := ( λ , λ , . . . , λ r ) ∈ Z r | λ ≥ λ ≥ · · · ≥ λ r } . Clearly, P r := P r ∩ N r is the set of the usual partitions of length at most r , namely the non–increasing sequences of non–negative integers with at most r non zero parts. If i > · · · > i k is a9ecreeasing sequence of integers, there exists one and only one bilateral partition µ ∈ P k such that i j = k − j + µ j . Therefore ([ b ] k µ ) µ ∈P k and ([ β ] l ν ) ν ∈P l where:[ b ] k µ = b k − µ ∧ . . . ∧ b µ k and [ β ] l ν = β l − ν ∧ . . . ∧ β ν l , are Q -bases of V k V and V l V ∗ respectively. Let z k := ( z , . . . , z k ) and w − k := ( w − , . . . , w − k ) betwo ordered finite sequences of formal variables. The V k V -valued formal power series b ( z k ) ∧ · · · ∧ b ( z )vanishes whenever z i = z j , for all 1 ≤ i < j ≤ k . Therefore it is divisible by the Vandermondedeterminant ∆ ( z k ) = Q < ≤ i 0, then s λ ( z k ) = s − λ ( z − k ) z k − · · · z k − k . (31)where − λ = ( − λ k , − λ k − , . . . , − λ ). If λ > λ k < 0, instead s λ k ( z k ) = s ( λ + λ k , ··· ,λ k − + λ k , ( z k ) Q kj =0 z λ k j . (32)It is then clear that all s λ ( z ), where λ runs on P k , are Q -linearly independent. The same holdstrue for ∆ ( w − l ). 10 .3 Let β ∈ V ∗ . The contraction β y : V V → V V can be depicted via the following diagram: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) β ( b r − λ ) β ( b r − λ ) . . . β ( b λ r ) b r − λ b r − λ . . . b λ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (33)to be read as follows. The scalar β ( b r − j + λ j ) is the coefficient of the element of V r − V obtained byremoving the j -th exterior factor from [ b ] r λ .The contraction of V r V against [ β ] l ν ∈ V l V ∗ is well defined as well. It is an element of V r − l V which can be represented as (See [1]):[ β ] l ν y [ b ] r λ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) β l − ν ( b r − λ ) . . . β l − ν ( b λ r )... . . . ... β ν l ( b r − λ ) . . . β ν l ( b λ r ) b r − λ . . . b λ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (34)to be read as follows. The Laplace-like expansion of the array (34) along the first row is analternating linear combination of contractions of elements of V k − V against elements of V l − V ∗ .Having already set the case k = 1 in (33), we have described it completely. Although it may be easily guessed, let us now make precise the definition of the contractionof an element of F against an element of V l V ∗ . Giving the definition on bases elements [ b ] m + λ of F and [ β ] l ν ( ν := ( ν ≥ . . . ≥ ν l ) of V l V ∗ will suffice. Let r ≥ ℓ ( λ ) ≤ r and ν l ≥ m − r and define: [ β ] l ν y [ b ] m + λ := ([ β ] l ν y [ b ] rm + λ ) ∧ [ b ] m − r . It is straightforward to see that the definition does not depend on the choice of the non-negativeinteger r > ℓ ( λ ). Let E ( z k , w − l ) = [ b ] k ( z k ) ⊗ [ β ] l ( w − l ) = X µ , ν [ b ] k µ ⊗ [ β ] l ν s µ ( z k ) s ν ( w − l ) , (35)be the generating function of the basis of V k V ⊗ V l V ∗ . It defines two maps E f ( z k , w − l ) : F → F J z k , w l , z − k , w − l ] (36)and E b ( z k , w − l ) := B ( ξ ) → B ( ξ ) J z k , w l , z − k , w − l ] (37)which we distinguish by putting a subscript in the notation and satisfying the compatibility relationimposed by the boson-fermion correspondence. More precisely we define: E f ( z k , w − l )[ b ] m + λ := [ b ] k ( z k ) ∧ [ β ] l ( w − l ) y [ b ] m + λ (38)and (cid:0) E b ( z k , w − l ) ξ m S λ ( x ) (cid:1) [ b ] = E f ( z k , w − l )[ b ] m + λ (39)where we have used the notation of (27) and (29).11 .6 Products of Schubert derivations. To further elaborate the shape of (38) and (39), weneed to introduce the following new piece of notation. Let σ + ( z k ) = σ + ( z ) · · · σ + ( z k ) , σ + ( z k ) = σ + ( z ) · · · σ + ( z k ) , (40)and σ − ( w l ) = σ − ( w ) · · · σ − ( w l ) , σ − ( z l ) = σ − ( w ) · · · σ − ( w l ) . (41)Equalities (40) and (41) must be read in End Q ( V V ) J z k K and End Q ( V V ) J w − l K respectively. Theyare multivariate HS-derivations of V V in the following sense: i) they are multi-variate becauseare End Q ( V V ) formal power series in more than one indeterminate, namely z k := ( z , . . . , z k ) and w − l := ( w − , . . . , w − l ), and ii) are HS derivations , being compatible with the wedge product: σ ± ( z k )( u ∧ v ) = σ ± ( z k ) u ∧ σ ± ( z k ) v and σ ± ( z k )( u ∧ v ) = σ ± ( z k ) u ∧ σ ± ( z k ) v . The following equality holds: β ( w − ) ∧ · · · ∧ β ( w − l ) y [ b ] m + λ = ∆ ( w − l ) Q lj =1 w m − l +1 j σ + ( w l ) σ − ( w l )[ b ] m − l + λ . (42) Proof. If l = 1 formula (43) reads as β ( w − ) y [ b ] m + λ = w − m σ + ( w ) σ − ( w )[ b ] m − λ and this is precisely [8, Proposition 6.13]. Assume the formula holds for l − ≥ 0. For notationalsimplicity let w l \ w := ( w , . . . , w l ) and w − l \ w − := ( w − , . . . , w − l ). Then β ( w − ) ∧ · · · ∧ β ( w − l ) y [ b ] m + λ = β ( w − ) y (cid:0) β ( w − ) · · · ∧ β ( w − l ) y [ b ] m + λ (cid:1) (Associativity of ” ∧ ” )= β ( w − ) y ∆ ( w − l \ w − ) Q lj =2 w m − l +2 j ! σ + ( w l \ w ) σ − ( w l \ w )[ b ] m − l +1+ λ = w − m + l − ∆ ( w − l \ w − ) Q lj =2 w m − l +2 j σ + ( w ) σ − ( w ) σ + ( w k \ w ) σ − ( w k \ w )[ b ] m − k + λ Now we use the commutation rule: σ − ( w ) σ + ( w k \ w ) = σ + ( w k \ w ) σ − ( w ) (cid:18) − w w (cid:19) · · · (cid:18) − w l w (cid:19) From which = w − m + l − w l − ( w − w ) · · · ( w − w l ) ∆ ( w − l \ w − ) Q w m + l − j · σ + ( w l ) σ − ( w l )[ b ] m − k + λ = w − m w l − w · · · w l Y (cid:18) w j − w (cid:19) ∆ ( w − l \ w − ) Q w m + l − j σ + ( w l ) σ − ( w l )[ b ] m − k + λ = ∆ ( w − l ) Q lj =1 w m − l +1 j σ + ( w l ) σ − ( w l )[ b ] m − k + λ 12s desired. The generating function (28) acts on F according to: X ν ∈P l [ β ] l ν s ν ( w − l ) y [ b ] m + λ = l Y j =1 w − m + l − j σ + ( w l ) σ − ( w l )[ b ] m − l + λ . (43) Proof. It is a consequence of equality (29) and of Proposition 2.7 up to dividing by the Vander-monde determinant. For all k ≥ : b ( z k ) ∧ · · · ∧ b ( z ) ∧ [ b ] m + λ = k Y j =1 z m +1 j ∆ ( z k ) σ + ( z k ) σ − ( z − k )[ b ] m + k + λ . Proof. By induction on k ≥ . If k = 1, the formula reads as b ( z ) ∧ [ b ] m + λ = z m +11 σ + ( z ) σ − ( z )[ b ] m +1+ λ and this is Proposition 6 . k − ≥ 0. Then, b ( z k ) ∧ · · · ∧ b ( z ) ∧ [ b ] m + λ = b ( z k ) ∧ ( b ( z k − ) ∧ · · · ∧ b ( z ) ∧ [ b ] m + λ ) (Associativity of ” ∧ ” )= b ( z k ) ∧ z m +1 k − · · · z m +11 σ + ( z k − ) σ − ( z k − )[ b ] m + k − λ · ∆ ( z k − )= z m + kk z m +1 k − · · · z m +11 ∆ ( z k − ) σ + ( z k ) σ − ( z k ) σ + ( z k − ) σ − ( z k − )[ b ] m + k + λ = z m + k +1 k Q k − j =1 z m +1 j Q k − j =1 (cid:18) − z j z k (cid:19) ∆ ( z k − ) ·· σ + ( z ) σ + ( z k − ) σ − ( z ) σ − ( z k − )[ b ] m + k − λ = z m + k +1 k z k − k k − Y j =1 z m +1 j k − Y j =1 ( z k − z j )∆ ( z k − ) · σ + ( z k ) σ − ( z k )[ b ] m + k − λ = k Y j =1 z m +1 j ∆ ( z k ) σ + ( z k ) σ − ( z k )[ b ] m + k + λ as desired. The generating function (26) acts on on the basis element [ b ] m + λ ∈ F accordingto: X µ ∈P k [ b ] k µ s µ ( z k ) ∧ [ b ] m + λ = k Y j =1 z m +1 j σ + ( z k ) σ − ( z k )[ b ] m + k + λ (44) Proof. By Proposition 2.9, using expression (26), dividing by the Vandermonde ∆ ( z k ).13 Fermionic and Bosonic Vertex Representation of gl ( V V ) . The following commutation rules holds in End Q ( F )[ z − , w K σ − ( z ) σ + ( w ) = (cid:16) − wz (cid:17) − σ + ( w ) σ − ( z ) (45)= exp X n ≥ n w n z n σ + ( w ) σ − ( z ) (46) Proof. Formula (45) is [8, Proposition 8.4, Formula (54)] and (46) uses the equality of formal powerseries (1 − x ) − = exp( P n ≥ x n /n ). Let p n ( z − k ) = k X i =1 z − ni and p n ( w l ) = l X j =1 w nj (the symmetric power sums New-ton polynomials). The following equalities holds on End Q ( ξ ) B ( ξ ) : σ − ( z k ) = k Y j =1 σ − ( z j ) = exp − X n ≥ n p n ( z − k ) ∂∂x n . (47) and σ − ( w l ) = l Y j =1 σ − ( z j ) = exp X n ≥ n p n ( w − l ) ∂∂x n (48) Therefore σ − ( z k ) σ − ( w l )[ b ] m + λ = exp − X n ≥ n ( p n ( z − k ) − p n ( w − l )) ∂∂x n ξ m S λ ( x ) [ b ] . (49) Proof. The operators X n ≥ n z ni ∂∂x n , X n ≥ n z nj ∂∂x n , X n ≥ n w np ∂∂x n , X n ≥ n w nq ∂∂x n commute for all choices of 1 ≤ i, j ≤ k and 1 ≤ p, q ≤ l . Then the product of their exponential isthe exponentials of their sum: σ − ( z k ) = k Y j =1 σ − ( z j ) = k Y j =1 exp − X n ≥ nz nj ∂∂x n = exp − X n ≥ n (cid:18) z n + · · · + 1 z nk (cid:19) ∂∂x n = exp − X n ≥ n p n ( z − k ) ∂∂x n , X n ≥ n p n ( z − k ) ∂∂x n and X n ≥ n p n ( w − l ) ∂∂x n commute.Thus: σ − ( z k ) σ − ( w l ) = exp − X n ≥ n p n ( z − k ) ∂∂x n exp X n ≥ n p n ( w − l ) ∂∂x n = exp − X n ≥ n ( p n ( z − k ) − p n ( w − l )) ∂∂x n . (50) The following commutation rules holds: σ − ( z k ) σ + ( w l ) = exp X n ≥ n p n ( w l ) p n ( z − k ) σ + ( w l ) σ − ( z k ) . (51) Proof. We first prove that σ − ( z k ) σ + ( w l ) = k Y i =1 l Y j =1 (cid:18) − w j z i (cid:19) − σ + ( w l ) σ − ( z k ) (52)For k = l = 1 the formula is Proposition (3.1). Suppose it holds for k − ≥ l = 1. Then σ − ( z k ) σ + ( w ) = k Y i =1 σ − ( z i ) · σ + ( w ) (definition of σ + ( z k ))= (cid:18) − w z k (cid:19) − k − Y i =1 σ − ( z i ) σ + ( w ) σ − ( z k ) (first step of induction on l )= (cid:18) − w z k (cid:19) − k − Y i =1 (cid:18) − w z i (cid:19) − σ + ( w ) k − Y i =1 σ − ( z i ) σ − ( z k ) (inductive hypothesison k )= k Y i =1 (cid:18) − w z i (cid:19) − σ + ( w ) σ − ( z k ) (definition of σ − ( z k )).Suppose now that (52) holds for all k ≥ l − ≥ 0. Then15 − ( z k ) σ + ( w l ) = σ − ( z k ) · σ + ( w l ) σ − ( w l − )= k Y i =1 (cid:18) − w l z i (cid:19) − σ + ( w l ) σ − ( z k ) σ + ( w l − )= l Y j =1 (cid:18) − w l z i (cid:19) − Y ≤ i ≤ k ≤ j ≤ l − (cid:18) − w j z i (cid:19) − σ + ( w l ) σ + ( w l − ) σ − ( z k )= Y ≤ i ≤ k ≤ j ≤ l (cid:18) − w j z i (cid:19) − σ + ( w l ) σ − ( z k ),which is precisely (52). To phrase (52) in the form (51) one first notice that (cid:18) − w j z i (cid:19) − = exp X n ≥ n w nj z ni . By a simple manipulation one sees that Y ≤ i ≤ k ≤ j ≤ l (cid:18) − w j z i (cid:19) − = Y ≤ i ≤ k ≤ j ≤ l exp X n ≥ n w nj z ni = exp X n ≥ n p n ( w l ) p n ( z − k ) as desired. Let R f ( z k , w − l ) : F → F [ z ± k , w ± l ] defined on homogeneous elements as: R f ( z k , w − l )[ b ] m + λ = Q ki =1 z m − l +1 i Q lj =1 w m − l +1 j ξ k − l [ b ] m + λ and R b ( z k , w − l ) ∈ Hom Q [ ξ ] ( B ( ξ ) , [ z ± k , w ± l ]) defined by( R b ( z k , w − l ) ξ m S λ ( x ))[ b ] = R f ( z k , w − l )[ b ] m + λ from which R b ( z k , w − l ) · Q ki =1 z m − l +1 i Q lj =1 w m − l +1 j ξ k − l The map R f ( z k , w − l ) commutes with Schubert derivations, in the sense that σ ± ( z k ) R f ( z k , w − l ) = R f ( z k , w − l ) σ ± ( z k ) and σ ± ( z k ) R f ( z k , w − l ) = R f ( z k , w − l ) σ ± ( z k ) . Proof. It is enough to prove that it commutes with σ ± i and σ ± j , i, j ≥ 0, which are by definition Q [ x k , w − l ]-linear. First of all recall that the product σ ± i [ b ] m + λ ( λ ∈ P r ) is ruled by some Pieri’sformulas σ ± i [ b ] m + λ = X µ ∈ P ± [ b ] m + µ , P + (resp. P − ) is the set of all partitions µ ≥ µ ≥ · · · ≥ µ r ( r ≥ ℓ ( λ )) such that µ ≥ λ ≥ · · · ≥ µ k ≥ λ k and | µ | = | λ | + i (resp. λ ≥ µ ≥ λ ≥ µ ≥ · · · ≥ λ r ≥ µ r and | µ | = | λ | − i ). Then we have σ ± i R f ( z k , w − l )[ b ] m + λ = σ ± i Y ≤ i ≤ k ≤ j ≤ l z m + l − i w m − l +1 j ξ k − l [ b ] m + λ = Y ≤ i ≤ k ≤ j ≤ l z m + l − i w m − l +1 j ξ k − l σ ± i [ b ] m + λ = Y ≤ i ≤ k ≤ j ≤ l z m + l − i w m − l +1 j ξ k − l X µ ∈ P ± [ b ] m + µ = R f ( z k , w − l ) X µ ∈ P ± [ b ] m + µ = R f ( z k , w − l ) σ ± i [ b ] m + λ Thus σ ± ( z k ) commutes with R f ( z k , z − l ) and so do σ ± ( z k ). Indeed: σ ± ( z k ) R f ( z k , z − l ) = σ ± ( z k ) R ( z k , z − l ) σ ± ( z k ) σ ± ( z k )= σ ± ( z k ) σ ± ( z k ) R ( z k , w − l ) σ ± ( z k )= R f ( z k , w − l ) σ ± ( z k ) . Notation as in (38) and (39) . Then: E f ( z k , w − l ) = exp X n ≥ n p n ( w l ) p n ( z − k ) Γ f ( z k , w l ) (53) and E b ( z k , w − l ) = exp X n ≥ n p n ( w l ) p n ( z − k ) Γ b ( z k , w l . ) (54) where the fermionic and bosonic vertex operators are, respectively Γ f ( z k , w l ) = R f ( z k , w − l ) σ + ( z k ) σ + ( w l ) σ − ( z k ) σ − ( w l )= R f ( z k , w − l ) exp X n ≥ x n ( p n ( z k ) − p n ( w l )) σ − ( z k ) σ − ( w l ) . (55) and Γ b ( z k , w l ) = R b ( z k , w − l ) exp X n ≥ x n ( p n ( z k ) − p n ( w l )) exp − X n ≥ p n ( z − k ) − p n ( w − l ) n ∂∂x n (56) Proof. We have: 17 f ( z k , w − l ))[ b ] m + λ = [ b ] k ( z k ) ∧ [ β ] l ( w − l ) y [ b ] m + λ (definition of E f ( z k , w − l ))= [ b ] k ( z k ) ∧ l Y j =1 w − m + l − j σ + ( w l ) σ − ( w − l )[ b ] m − l + λ (Corollary 2.8)= Q ki =1 z m − l +1 i Q lj =1 w m − l +1 j σ + ( z k ) σ − ( z k ) σ + ( w l ) σ − ( w − l )[ b ] m + k − l + λ (Corollary 2.10)= R ( z k , w − l ) σ + ( z k ) σ − ( z k ) σ + ( w l ) σ − ( w − l )[ b ] m + λ (Definitionof R ( z k , w − l ))By invoking the commutation relation proven in Proposition 3.3, one obtains E f ( z k , w − l )[ b ] m + λ = exp X n ≥ n p n ( w l ) p n ( z − k ) R ( z k , w − dir ol ) σ + ( z k ) σ + ( w l ) σ − ( z k ) σ − ( w l )[ b ] m + λ (57)which already prove that the expression of E f ( z k , w − l ) is precisely (53). To continue with, the B ( ξ )-module structure of F says that F m is an eigenspace of σ + ( z k ) σ + ( w l ) with eigenvalue k Y i =1 exp X n ≥ x n z ni l Y j =1 exp − X n ≥ x n w nj = exp X n ≥ x n p n ( z k ) exp − X n ≥ x n p n ( w l ) = exp X n ≥ x n ( p n ( z k ) − p n ( w l )) . (58)Thus formula (57), up to replacing σ + ( z k ) σ + ( w l ) by its eigenvalue (58) with respect to F , isprecisely (53) with Γ f ( z k , w l ) given by expression (55). To prove (54) we recall that( E b ( z k , w − l ) ξ m S λ ( x ))[ b ] = E f ( z k , w − l )[ b ] m + λ = exp X n ≥ n p n ( w l ) p n ( z − k ) Γ f ( z k , w l )[ b ] m + λ Now Γ f ( z k , w l )[ b ] m + λ = R f ( z k , w − l ) exp X n ≥ x n ( p n ( z k ) − p n ( w l )) σ − ( z k ) σ − ( w l )[ b ] m + λ However, by Proposition 3.2, formula (49), σ − ( z k ) σ − ( w l )[ b ] m + λ = exp − X n ≥ p n ( z − k ) − p n ( w − l ) n ∂∂x n ξ m S λ ( x ) [ b ] . which shows that Γ f ( z k , w l )[ b ] m + λ = (Γ b ( z k , w − l ) ξ m S λ ( x ))[ b ] proving the theorem. 18 .7 Remark. In formula (54) let us set k = l = 1 and call z = z and w = w . Then p n ( z ± ) = z ± n and p n ( w ± ) = w ± n . Then E b ( z, w ) = exp X n ≥ n w n z n Γ b ( z, w )where Γ b ( z, w ) = R b ( z, w ) exp( X n ≥ x n ( z n − w n )) exp − X n ≥ z − n − w − n n ∂∂x n Keeping into account that exp X n ≥ n w n z n = 11 − wz and using the definition of R b ( z, w − ) one sees that E b ( z, w ) | B ( m ) = z m w m − wz exp( X n ≥ x n ( z n − w n )) exp − X n ≥ z − n − w − n n ∂∂x n which is the celebrated DJKM formula. Acknowledgments. This work is one of the topics touched in the Ph.D. thesis of the first author,mostly redacted during her hosting at the Department of Mathematical Sciences of Politecnico ofTorino under the sponsorship of Ministry of Science of the Islamic Republic of Iran. The secondauthor profited of the support of Finanziamento Diffuso della Ricerca (no. 53 RBA17GATLET) andProgetto di Eccellenza del Dipartimento di Scienze Matematiche, 2018–2022, no. 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