Polynomial ring representations of endomorphisms of exterior powers

An explicit description of the ring of the rational polynomials in r indeterminates as a representation of the Lie algebra of the endomorphisms of the k-th exterior power of a countably infinite-dimensional vector space is given. Our description is based on results by Laksov and Throup concerning the symmetric structure of the exterior power of a polynomial ring. Our results are based on approximate versions of the vertex operators occurring in the celebrated bosonic vertex representation, due to Date, Jimbo, Kashiwara and Miwa, of the Lie algebra of all matrices of infinite size, whose entries are all zero but finitely many.


Statement of the main result
The purpose of this paper is to supply an explicit description of the polynomial ring B r ∶= ℚ[e 1 , … , e r ] as a module over the Lie algebra of endomorphisms of k-th exterior powers of a vector space V ∶= ⨁ i≥0 ℚ ⋅ b i of infinite countable dimension. Let j ∶ V → ℚ be the unique linear form such that j (b i ) = ji so that V * ∶= ⨁ j≥0 ℚ ⋅ j is the restricted dual of V. Write the r-th exterior powers of V and V * , respectively, as where and range over the set P r of all the partitions of length at most r.
A strict relative of ⋀ r V is the vector space B r , which can be identified with the ring of symmetric polynomials in r indeterminates. It is well known that it possesses a ℚ -basis formed by certain Schur determinants S ∶= S (e 1 , … , e r ) (Cf. Sect. 2.4, adopting the notation of [12, p. 41] For all k, r ≥ 0 , we consider the B r -representation of gl( ⋀ k V) , which we understand as the action: where [ ] k ⌟ ∶ ⋀ r V → ⋀ r−k V is the standard contraction operator (Sect. 2.2).
To express the gl( ⋀ k V)-action (1) on B r through a compact formula, a standard philosophy suggests to use generating functions. To this purpose, let us introduce some notation. Let k ∶= (z 1 , … , z k ) and k ∶= (w 1 , … , w k ) be two sets of formal variables. The k-tuples of the formal inverses (z −1 1 , … , z −1 k ) and (w −1 1 , … , w −1 k ) will be denoted by −1 k and −1 k respectively. If ∶= (u 1 , … , u k ) are arbitrary formal variables, denote by p i ( k ) the power sum u i 1 + ⋯ + u i k of degree i. The standard notation s ( k ) and s ( −1 k ) stands for the symmetric Schur polynomials in the k and −1 k (See [12, p. 40]). Let E r (z) ∶= 1 − e 1 z + ⋯ + (−1) r e r z r ∈ B r [z] , set by convention b j = 0 if j < 0 and denote by −1 the locally nilpotent endomorphism of V mapping b j ↦ b j−1 for all j ≥ 0 . Let ∶ End(V) ↦ End( ⋀ V) be the natural representation of End(V) as a Lie algebra of (even) derivations of ⋀ V.
The main ingredients to state our main result are certain vertex operators −1 k ] acting on the exterior algebra ⋀ V . They are introduced in Definition 4.3 as products of Schubert derivations, and studied in more detail only in Sects. 6 and 7. However, if r is big with respect to the length of the partition they can be explicitly written as (1) and Consider now the generating formal power series defined by the equality: Main Theorem. For all k, r ≥ 0 and all ∈ P r , the action of E( k , −1 k ) on the basis element S of B r is given by: The above result corresponds to Theorem 8.5 within the text and supplies the explicit description of the ring B r as a module over the Lie algebra gl( ⋀ k V) , for all k ≥ 0 . In fact, the E k , -image of S is determined by the coefficient of s ( k )s ( −1 k ) obtained by the expansion of the right hand side of (2). This may sounds tricky to evaluate, but is nothing else than the coefficient of of the right hand side of (2), multiplied by the Vandermonde determinants of k and −1 k . That B r is a representation of gl( ⋀ k V) is easy to see in very special cases. For k = 0 , it is the multiplication by rational numbers, as ⋀ 0 V = ℚ , while for k > r is the trivial null representation. The case r = k = 1 recovers the well known general fact that every vector space is a module over the Lie algebra of its own endomorphisms. In fact the linear extension of the set map e i 1 ↦ b i is a vector space isomorphism B 1 → V , making B 1 into a gl(V)module, by pulling back that structure from V. Our Main Theorem then takes into account the general case.

The boson-fermion correspondence and the DJKM representation
The gl( ⋀ k V)-module structure of B r , described in Main Theorem, will be referred to as bosonic representation of gl( ⋀ k V) , by a possibly strong, but suggestive, abuse of terminology, due to the evident relationship with the pioneering work by Date, Jimbo, Kashiwara and Miwa (DJKM) [8] (see also [24,25]) which also fits into the more general framework considered in the reference [7].
As a matter of fact, one main motivation of this paper was to better understand a fundamental, although elementary, representation theoretical fact. Let V ∶= ⨁ j∈ℤ ℚ ⋅ b j be a vector space with basis ∶= (b j ) j∈ℤ , parameterized by the integers (one may think of V as being the vector space ℚ[X −1 , X] of the Laurent polynomials) and V * its restricted dual with basis ( j ) j∈ℤ . It is well known that V ⊕ V * supports a canonical structure of Clifford algebra C ∶= C(V ⊕ V * ) ([9, p. 85] or [18]) and that the Fermionic Fock space F (also called the semi-infinite wedge power and denoted by ⋀ ∞∕2 V ) is an irreducible representation of C .
More precisely, F is an invertible module over the Lie super-algebra C generated by a distinguished vector �0⟩ , the vacuum, that in the formalism of the infinite wedge power can be suggestively written The huge Clifford algebra C , whose elements are finite linear combinations of words of the form b i i ⋯ b i h j 1 ⋯ j k , contains in a natural way all, but not only, the Lie algebras gl( ⋀ k V) , for all k ≥ 0 . In particular, it turns out that F is a gl( Our paper, however, aims to look at more traditional, but relevant, contexts. Exactly as in the case of the Fermionic Fock space F , the exterior algebra of V ≅ ℚ[X] is an irreducible representation of the canonical Clifford algebra C supported on V ⊕ V * . This occurrence convinced ourselves to give a closer look to the gl( ⋀ V)-structure of ⋀ V , certainly not treated in any literature we have consulted up to now. We have so gotten a description of the gl( ⋀ k V)-module structure of B r , which generalises the case r < ∞ and k = 1 studied in [20]. The output is that the direct sum ⨁ k≥0 gl( ⋀ k V) is a Lie subalgebra of gl( ⋀ V) , represented by B r for all r ≥ 0 . In the case of the fermionic Fock space, the gl( ⋀ 1 V)-structure of B ∞ is the DJKM one [8,24]. The general case, which amounts to the description of the gl( ⋀ V)-module structure of F , is faced in the contribution [2] as a best example of the extension of the techniques used in [21].

Methods and their applications
The vertex operators occurring in our description of B r as a representation of gl( ⋀ k V) are defined by means of Schubert derivations, which are distinguished Hasse-Schmidt (HS) derivations on exterior algebras. HS derivations were first introduced in [13] and extensively treated in [18]; see also the survey [1] or [6, p. 116], for more discussions. In a finite dimensional context Schubert derivations are related to Chern and Segre polynomials of the tautological bundle over a Grassmannian. The point is that the Segre and Chern polynomials act as a HS-derivation on the exterior algebra of the homology of the projective space, which is the same as saying that to do Schubert calculus on Grassmannians, Bézout theorem suffices.
The Schubert derivations we introduce here, denoted by + (z), + (z) , − (w) and − (w) enjoy some nice commutation rules. Those with the same sign as subscripts commute in the algebra of endomorphisms of the exterior algebra. However, due to the fact that − (w) and − (w) are locally nilpotent, they commute with + (z) and + (z) only up to the multiplication by a rational function.

Organization of the paper
To be as much self-contained as possible, we collect most of preliminaries and basic notation in Sect. 2. The first part recalls basics of the theory of symmetric polynomials as, e.g., in [12]. The second part accounts for the invertible Hasse-Schmidt derivation on an exterior algebra, essential in the subsequent sections.
Section 3 contains the explicit expression of the Schubert derivations that makes evident their strong connection with vertex operators. Section 4 also contains an effective definition of what we have proposed to name vertex operators on a Grassmann algebra, because an obvious relationship with those occurring in the classical boson-fermion correspondence. Also, to check the Main Theorem without neglecting any minimum detail, we state and prove in Sect. 4 relevant commutation rules, some of which can be recognized within the phrasing of the categorical framework for the boson-fermion correspondence, depicted in [10] (see also [28] for a recent update).
In Sect. 5 we instead study commutation rules involving the contraction operator: this is a typical issue in the situation involving a finite wedge power. In fact, the finiteness makes the subject trickier than when working with the infinite wedge power.
Vertex operators in the sense of Definition 4.3 are homogeneous operators on the exterior algebra, one of positive and the other of negative degree. We devote one section to each one of them (Sects. 7 and 8) to dig up their relationship with basic computations in multilinear algebra, such as wedging and contracting.
Certainly this idea is already present in the infinite wedge power context (e.g. [25,Chapter 5]), but the present article, together with [19][20][21], is the first instance of applications of the techniques and ideas in finite dimensional landscapes.
Finally, last Sect. 8 is concerned with the proof of the Main Theorem together with some of its straightforward declinations in terms of certain familiar objects, like suitable deformations of the same Giambelli's determinants occurring in classical Schubert Calculus, see Theorem 8.10. To achieve the proof of the Main Theorem, some preliminary lemmas (such as 8.2 and 8.3) are proved. We believe that these lemmas along with Theorems 6.5 and 7.3, are interesting in their own, as pieces of multilinear algebra properties addressed to wider general mathematical audiences.

Partitions
A partition is a monotonic non increasing sequence ∶= ( 1 ≥ 2 ≥ … ) of non negative integers, said to be its parts. The length ( ) is the number of its non zero parts, and � � = ∑ i≥0 i is its weight. We denote by P r be the set of all partitions of length at most r.

Exterior powers, exterior algebras and duality pairing
Denote by (z) and (w −1 ) the generating series of the basis elements of V and of V * respectively, i.e.: The The algebra structure is given by the ℚ-linear extension of the juxtaposition. To each ∈ P r we associate

The ring B r
Let r ≥ 1 . The main character of this paper is the polynomial ring Given the generic polynomial E r (z) ∶= 1 − e 1 z + ⋯ + (−1) r e r z r ∈ B r [z] , one considers the sequence H r ∶= (h j ) j∈ℤ defined by the equality: holding in B r [ [z] ] . In particular h j = 0 if j < 0 and h 0 = 1 . Moreover for j ≥ 0 , the term h j is an explicit polynomial in (e 1 , … , e r ) , homogeneous of degree j, once one gives weight i to e i . The Schur determinants .
form a ℚ-basis of B r parametrized by the partitions of length at most r: It follows that B r is naturally isomorphic to ⋀ r V via the ℚ-linear extension of the sets map

Schur polynomials
Especially in the last section we shall be concerned with Schur polynomials in a set of indeterminates. We recall them here. For each partition of length at most k and any set of k formal variables k ∶= (x 1 , … , x k ) , one defines This is an skew symmetric polynomials in (x 1 , … , x k ) and therefore divisible by the Vandermonde determinant The Schur polynomial associated to k and the partition is defined by the equality often said to be the Jacobi-Trudy formula.

Hasse-Schmidt derivations on exterior algebras
] denote the formal power series in the indeterminate z with coefficients in the exterior algebra If S is any set of indeterminates over ℚ , denote by ℚ[S] the corresponding algebra of formal power series. The following is an extended reformulation of the main definition of the reference [13] (see also [18]). By a Hasse-Schmidt derivation on which, by abuse of notation, will be denoted by the same symbol (instead of the more precise, but lengthier, (12) is equivalent to the system of relations The notation will be used as a shorthand for the equality -valued formal power series and its inverse, D(z) , is an HS-derivation as well. (14) and (15) are implicitly assuming the ℚ[[z]]-linearity of D(z) we alluded to in Definition 2.5. The extension of the linearity of HS-derivations over polynomial algebras will be assumed in the following without any further mention.

Definition
The Schubert derivations on ⋀ V are the HS-derivations

Remark
It is easily seen that ± (z) and ± (z) are the unique HS-derivations on ⋀ V such that and putting b i = 0 for i < 0.

B r -module structure of ⋀ r V
We exploit the Schubert derivation + (z) or, equivalently, its inverse + (z) , to endow ⋀ r V with a B r -module structure, by declaring that e i u = i u or, equivalently, h i = i , for all ∈ ⋀ V . In particular: The fact that such a product structure is compatible with the natural vector space isomorphism B r → ⋀ r V given by (11) The fact that ⋀ r V is a free B r -module of rank 1 generated by [ ] r 0 , as prescribed by equality (22), shows that the Schubert derivations − (z), − (z) induce maps B r → B r [z −1 ] which, abusing notation, will be denoted in the same way. Their action on a basis element Δ (H r ) of B r is defined through its action on ⋀ r V:

Product of Schubert derivations
For k ≥ 1 , let k denote the ordered k-tuple (z 1 , … , z k ) of formal variables. By The maps occurring in formulas (27) are multivariate HS derivations on ⋀ V , in the sense that, for instance, + ( k )(u ∧ v) = + ( k )u ∧ + ( k )v , as it is easy to check and adopting the linear extension of the Schubert derivation to polynomial coefficients according to Definition 2.5. The same holds verbatim for − ( k ) and ± ( k ) . It is an important point that the multivariate HS derivations in (27) are symmetric in the formal variables z i and w i . This is a consequence of the first of the commutation rules obeyed by the product of Schubert derivations listed in this section and to be used in the sequel.

Proposition
Let z, w be arbitrary formal variables. The equalities Proof Formulas (28) and (29) are obvious consequences of the fact that if i, j ≥ 0 , then ±i and ±j are pairwise commuting. It is sufficient, then, to show that they commute when restricted to V, because if they do, then (27) ± ( k ) ∶= ± (z 1 ) ⋯ ± (z k ) and ± ( k ) ∶= ± (z 1 ) ⋯ ± (z k ).
(28) ± (z) ± (w) = ± (w) ± (z), (29) ± (z) ± (w) = ± (w) ± (z), ± (z) ± (w)[ ] r = ± (z) ± (w) r = ± (w) ± (z) r , having used notation as in (13). But ±i ±j u = i+j ±1 u = ±j ±i u for all u ∈ V , and then the claim follows. ◻ In order to give a compact expression of the gl( ⋀ k V)-module structure of B r , we shall need to introduce a generalisation of the classical vertex operators arising in the context of the so-called boson-fermion correspondence, like in e.g. [25]. We look at it as a generalisation of the isomorphism B r → ⋀ r V , recalled in Sect. 2.1, reaffirmed and refined in Proposition 3.5.

Definition By vertex operators on
of degree 1 and −1 , with respect to the exterior algebra graduation, given by: Proposition 4.2 guarantees that the vertex operators Γ( k ) and Γ * ( k ) are symmetric in the formal variables (z 1 , … , z k ) . They will be studied in a more detailed way in Sects. 6 and 7, exploiting further commutation relations, for which we need the preliminary work exposed below. As a matter of fact, we notice that the commutativity of the product of Schubert derivations is granted only if they are of the same kind (both subscripts " + " or both subscripts "−"). In general, for i, j > 0 , i and −j do not commute, because −j is locally nilpotent. The simplest example is: The general pattern is that commutativity only holds up to the multiplication by a rational function.

Commutation rules for contractions
The goal of this section is to prove the following

Theorem For all u ∈
⋀ r V , the following commutation rule holds: To prove Theorem 5.1 some preparation is needed.

Diagrams for contractions
Let us begin to introduce a piece of useful notation. If ∈ V * , we represent the contraction as defined by equality (7), via the diagram: to be read as follows. The scalar (−1) j+1 ⌟u j ∶= (−1) j (u j ) is the coefficient of the element of ⋀ r−1 V obtained by removing the wedge factor u j from u 1 ∧ u 2 ∧ ⋯ ∧ u r . For example This is exactly the expanded expression of the contraction ⌟(u 1 ∧ u 2 ∧ u 3 ) . Recall now the generating function (w −1 ) ∶= ∑ j≥0 j w −j introduced in formula (3).

Proposition For all u ∈
⋀ r V: (40) = (w −1 )⌟b j − (w −1 )⌟b j+1 z (Action of (w −1 )⌟) By expressing the contraction via diagram (37), one has: which by (40) is equal to: Since the determinant occurring in (41) is a linear combination of + (z) is a HS derivation), where we denoted by (j) the partition of lenght at most r − 1 obtained by omitting the j-th part, it follows that the action of + (z) can be factorized from the bottom row of (41), giving which ends the proof of the Proposition. ◻  The last equality proves (42), due to the arbitrary choice of ∈ ⋀ r−1 V * ≅ ( ⋀ r−1 V) * .

Lemma
The operator + (z) commutes with contracting against 0 , i.e. for all ∈ P r Proof There are two cases. If ( ) = r , both members of (43) vanish. If ( ) ≤ r − 1 , then We are now in position to provide the

Proof of Theorem 5.1
We have: and Theorem 5.1 is thence proven. ◻

The vertex operator 0( k )
The main purpose of this section is to interpret the vertex operator Γ( k ) , introduced in Definition 4.3, formula (30), in terms of wedging operation on the exterior algebra. It will generalise [19,Proposition 4.2]. This will be achieved in Theorem 6.5 below and will be used in our Main Theorem 8.5.

Lemma
For all j ≥ 0 and all k ≥ 1 one has and where e i ( k ) and h i ( k ) are, respectively, the elementary and complete symmetric polynomial of degree i in the indeterminates k ∶= (z 1 , … , z k ). (44) is the content of [21,Lemma 5.7] to which we refer to. Formula (45) is a consequence of (44), keeping into account that + ( k ) and + ( k ) are mutually inverse in

Lemma One has:
where Proof The proof works the same as in [21,Lemma 6.7]. The formula is true for k = 1 , because By induction, suppose that (46) holds for k − 1 ≥ 0 . Then it holds for k. Indeed as desired. ◻

Lemma
The following equality holds for all 1 ≤ i ≤ k: Proof Recall the following definition of the elementary symmetric polynomials in k indeterminates through generating functions: By dividing both sides of (48) by e k ( k ) = z 1 … z k we get The claim then follows by comparing the coefficient of t i in in either side of (49). ◻

Lemma
For all k ≥ 1 , r ≥ 0 and ∈ P r : Proof Equality (50) holds for r = 1: Therefore the property is true for r = 1 . Assume now (50) holds true for r − 1 ≥ 0 . Then as claimed. ◻

Theorem
Proof Recall that we consider all the Schubert derivations extended by linearity over rings of formal power series with rational coefficients. See definition 2.5. Then our arbitrary u is intended as a linear combination of [ ] r with coefficients being polynomials. Then we can assume with no harm that u = [ ] r , a basis element of ⋀ r V . Keeping the same notation as in Sect. 2, we first apply integration by parts. Then (By integration by parts (14)) as desired. ◻ If k = 1 , and z = z 1 , one obtains which is precisely [19,Proposition 5.4] or [20,Proposition 3.2]. Their shape there looks more involved because of a different notation.

The vertex operator 0 * ( k )
In the same vein of Sect. 6, the present one will be devoted to interpret the action of the vertex operator Γ * ( k ) on ⋀ V in terms of contraction operators. The output will be Theorem 7.3, stated at the end of the section, another building block of the main Theorem 8.5. We begin with some preparation.

Lemma
The following equality holds for all r ≥ 1 and all ∈ P r : Proof This is [19,Lemma 5.8].

Theorem
The following equality holds: Proof For k = 1 the property is just Lemma 7.2. Arguing by induction, let us assume the claim holding true for 0 ≤ k − 1 ≤ r − 1 and let us show it holds for k. We have Using the inductive hypothesis: By applying Lemma 7.2, one gets: Now we use the commutation rules prescribed by Theorem 5.1 and Lemma 5.6: as claimed. ◻

The main theorem and its declinations
In this section we shall be concerned with the several declinations of the main theorem describing the B r representation of gl( ⋀ k V).

Preparation
Let gl( It is a Lie sub-algebra of the endomorphisms of ⋀ k V , with respect to the natural commutator. With the same notation as in 1.1, a basis of Then the gl( ⋀ k V)-module structure of B r is defined through the following equality holding in ⋀ r V: This action is very easy to describe in the case k = r , but it becomes trickier when r − k > 0 . To describe it we shall consider the generating function Our main result will consist in the explicit description of E( k , −1 k )Δ (H r ) in case k ≤ r (because otherwise one would obtain the trivial null action), where E( k , −1 k )Δ (H r ) is such that and where s ( k ) and s ( −1 k ) denote the Schur symmetric polynomials labeled by the partitions related to the variables k and −1 k respectively. Proof The one we propose consists in expanding the wedge product of the generating series of the basis ( j ) j≥0 of V * : (58)

Lemma
whence the claim, obtained by multiplying both (59) and (60) by ( ∏ k j=1 w k j )∕Δ 0 ( k ) . ◻ where in the last equality we have repeatedly used the module structure of ⋀ r V over B r .

Lemma
Then: and the result now follows from Theorem 6.5. ◻ We can finally express the action of the generating function E , ( k , −1 k ) on a basis element of B r . (First Version). For all ∈ P r :  x j z j = 1 E r (z)
where p j ( k ) = z j 1 + ⋯ + z j k is the i-th power sum symmetric polynomial in (z 1 , … , z k ) and where x i is precisely the i-th degree power sum in the r universal roots (y 1 , … , y r ) of the polynomial E r (z) , i.e. E r (z) = ∏ r i=1 (1 − y i z) in the universal splitting ℚ-algebra for the polynomial E r (z) ∈ ℚ[z] . This allows to shape our result in the form We owe a lot to the precious referee's comment which helped us to improve the presentation of our paper. For discussions and criticisms, however, we want to primarily thank Inna Scherbak, who first suggested us to generalise the DJKM picture in the way as now stands in the present paper. Finally, we are indebted to Joachim Kock and Andrea T. Ricolfi for carefully reading and many other kinds of assistance.

Corollary
Funding Open access funding provided by Politecnico di Torino within the CRUI-CARE Agreement.
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