A. N. Sergeev, E. D. Zharinov. Pieri Formulae and Specialisation of Super Jacobi Polynomials Pieri Formulae and Specialisation of Super Jacobi Polynomials

We give a new proof of the fact that the Euler supercharacters of the Lie superalgebra osp(2m + 1, 2n) can be obtained as a certain limit of the super Jacobi polynomials. The known proof was not direct one and it was mostly based on calculations. In this paper we propose more simple and more conceptional proof. The main idea is to use the Pieri formulae from the beginning. It turns out that the super Jacobi polynomials and their specialisations can be uniquely characterised by two properties. The first one is that they are eigenfunctions of CMS operator and the second one is that they satisfy the Pieri formulae. As by product we get some interesting identities involving a Young diagram and rational functions. We hope that our approach can be useful in many similar cases.


INTRODUCTION
Generalisations and specialisations of Macdonald polynomials play an important role in many fields of mathematics (see for example [1][2][3] and [4][5][6]). Super Jacobi polynomials are one of such generalisations (see [7]). They can be naturally connected to the theory of Lie superalgebras and deformed quantum Calogero -Moser -Sutherland (CMS) systems. Let be the family of super Jacobi polynomials in + indeterminates (see [7]) labeled by the set of partitions such that +1 . It has been proved in the paper [8] that lim where is the Euler supercharacter (with a special choice of the parabolic subalgebra) of the Lie superalgebra osp(2 + 1, 2 ) and , , are the parameters of the super Jacobi polynomial and the number ( ) is defined in Theorem 6.
This result is interesting and important from the representation theory point of view and from the point of view of CMS systems as well. But the proof given in the paper [8] is rather involved. It uses the infinite dimensional version of the Jacobi symmetric polynomials, Okounkov formula for Jacobi symmetric functions, generalisation of the Weyl type formula for super Schur polynomials and some other more technical means.
The goal of this paper is to give more simple and more conseptual proof of that result. In the present paper we use two main properties of super Jacobi polynomials. The first one is that they are eigenfunctions of the deformed Calogero -Moser -Sutherland operator and the second property is that they satisfy the Pieri identity. This two properties define them uniquely and the same is true for the Euler supercharacters as well. So instead of calculating the limit of the super Jacobi polynomials we calculate the limit of the CMS operator (which is trivial) and the limit of the coefficients of the Pieri formulae. The last limit can be calculated using some simple identities with rational functions and Young diagrams. On the other side we have an explicit formula for the Euler supercharacters obtained by V. Serganova [9]. Using this explicit formula we prove the corresponding Pieri formula and the fact that the Euler supercharacters are eigenfunctions of the limiting CMS operator. As a result we get the coincidence of the family of specialised super Jacobi polynomials and the Euler supercharacters. We believe that this method can be applied in many similar cases. For example using this approach it is easy to prove the Weyl type formula for super Schur polynomials.

SUPER JACOBI POLYNOMIALS
In this section we define super Jacobi polynomials using the fact that they satisfy the Pieri formula and that they are eigenfunctions of the deformed Calogero -Moser -Sutherland operator. The deformed CMS operator of type , has the following form (see [7, p. 1712 )︂ , where = , = and the parameters , , , , satisfy the relations = , 2 + 1 = (2 + 1). In the formulae below we always suppose that ℎ = − − − 1 2 − where , are non negative integer numbers. In order to define coefficients of the Pieri formulae let us introduce the flowing notations: let ( , ) be the set of partitions such that +1 and ( ) = 2 + 2 3 + . . . , = 2 ( ′ ) + 2 ( ) + | |(2ℎ + 2 + 1), where ′ denote the conjugate partition to and denote the box ( ) ( is the row index and is the column index) of the Young diagram which corresponding partition is (see [4]). Let us set .
Let us also denote by + ( ) the set of partitions which can be obtained from by adding one box and by − ( ) we will denote the set of partitions which can be obtained from by deleting one box. Let us also set ( ) = + ( ) ∪ − ( ) ∪ { }.
If ∈ − ( ) and = − 1 then we set , and , = ( ) ] be the algebra of Laurent polynomials in + indeterminates. Now we are ready to define super Jacobi polynomials.

Proof. Let
Then it is not difficult to verify that from the conditions = and , ∈ ( ) it follows that = . Therefore the operator is well defined. So if a family of polynomials { } satisfy the conditions of the Theorem then from the last formula in (1) it follows that and uniqueness can be proved by induction on the number of boxes in . Existence follows from [7, sec. 7].

Lemma 1. 1. Let be a Young diagram such that we can add a box to the i-th row. Then the following equalities hold true
2. Let be a Young diagram such that we can delete a box from the i-th row. Then the following equalities hold true Proof. Let us prove (3). If Therefore

Let us check that ∏︁
Indeed if +1 = then ∆ = ∅ and the left hand side of the equality is 1 by definition. The right hand side is also 1.
If +1 < then for any ∈ ∆ we have ′ = and equality (7) is easy to verify. So we have proved the first equality. The second equality follows from the first one by the substitution → − . Equalities (5), (6) can be proved in the same way. Now we are going to prove our main result of this section.

Theorem 2. The following statements hold true
Proof. Let us prove the statement 1). Let us represent ( ) as the product ( ) = ( , 1) ( , 2), where ( , 1) is the part of ( ) which does not depend on ℎ and ( , 2) is the part which depends on ℎ. The same notation ,1 (1), ,2 (1) will be used for (1). We are going to prove two equalities Let us prove the first one. It can be rewritten in the form: where ( ) is the added box. It is easy to check the following equality Since the box ( ) can be added to then we have = −1 and ′ = −1. Therefore Besides we have (˜+ ′ − + 3) = 1 and if we substitute =˜+ 1 in the equality (3) then we get Therefore we have proved the first equality in (8). By using the identity (4) we can prove in the same way the second equality in (8) and therefore we have proved the statement 1) of the Theorem.
The statement 2) can be proved analogously using the identities (5) and (6). Now let us prove the statement 3). Consider the expansion into partial fractions of the function (ℎ). If ∈ ( ) + and = + 1 then where is a polynomial. In the same way if ∈ ( ) − and = − 1 then It is not difficult to verify the following equalities Besides we have = ( , 1)(2ℎ + 2 + + 1) and it can be checked that This proves statement 3) and the Theorem. A proof of the Theorem easily follows from the Theorem 2.

SPECIALISATION OF THE SUPER JACOBI POLYNOMIALS
In the previous section we specialised the coefficients of the Pieri formula. Now we can prove an existence of specialised super Jacobi polynomials and the corresponding Pieri formulae.

Specialisation as → −1
Proof. Suppose that + 2 / ∈ Q then it is not difficult to verify that if˜=˜and , ∈ ( ) then = . Therefore there exists a limit Now let prove the existence of lim →−1 by induction on | |. If | | = 0, then ∅ = 1 and the existence is clear. Let | | > 0 and can be obtained from by deleting one box. Then from formula (2) and statement 1) of Theorem 2 it follows that lim →−1 =d oes exist and the following equality holds truẽ So the equality (9) can be obtained from (1) by taking the limit as → −1.
Specialisation as ( , ) → (−1, 0) Let us prove the existence lim ( , )→(−1,0)˜= ♯ . In this case the equation ♯ = ♯ can have more than one solution for , ∈ ( ). The following Lemma is not difficult to prove. So the previous Lemma shows that the equation ♯ = ♯ can have not more than two solutions for , ∈ ( ). For this case we need the following Lemma.

Lemma 3. Let
Let us denote the limit above by ℒ ♯ , . Now we can prove the main result of this section.
Let us take the limit as ( , ) → (−1, 0) in the equality (10). Then we get This proves the existence of in this case. Suppose now that the equation ♯ = ♯ has two different solutions , ∈ ( ). Then by Lemma 3 we have By Lemma 2 we have | | < | | therefore ♯ is well defined and we proved the existence. Uniquness follows from the formulae (12), (13).

EULER SUPERCHARACTERS
In this section we prove two main results using an explicit formula for Euler supercharacter by V. Serganova [9,10]. The first one is Pieri formulae and the second result is that Euler supercharacters are eigenfunctions of the deformed CMS operator. As a corollary for any partition ∈ ( , ) we get the coincidence (up to sign) of the Euler supercharacter and the specialised super Jacobi polynomial ♯ . In order to define Euler supercharacters let us introduce new variables = + −1 , = + −1 and the following polynomials in one variable where is integer.
It it easy to check that these polynomials satisfiy the following recurrence relations Let us also for ∈ ( , ) define = − and the following numbers The following formula for Euler supercharacters in variables , can be easily deduced from the last formula for Euler supecharacter in [8, p. 4307] where ( ) equals to the number of pairs ( , ) ∈ such that − + < 0.

Euler supecharacter
is an eigenfunction of the deformed CMS operator ℒ ♯ with eigenvalue ♯ .
Proof. By + (and the same for − ) we will denote the diagram obtaining from by adding one box to the -th row (or deleting one box from the -th row). The same notations ± will be used for columns. Let us also denote by the product Π ( , ) ( ) ( ). We are going to calculate { } and { }. The following equalities hold true All the above equalities except the equalities (17) can be easily proved from the definition of the Euler supercharacters. So let us prove the first equality in (17). It is easy to check that − ( ) = − ( ). Therefore it is enough to prove this equality in the case when is a row or a column and = . Let = ( ), > 0 be a row.
and we need to prove that Using the recurrence relations (14), (15) we can rewrite the previous equality in the form Now using the alternation operation over subgroup {1} × we can rewrite the previous equality in the form And in this form this equality can be easily checked. The other equalities in (17) can be proved in the same way.
It is known (see [11]) that ℒ ♯ −1 = ∑︀ Let us note that if = , then ( , , , ) = 0. So we see that it is enough to prove that is an eigenfunction of the operator ℒ ♯ −1 with eigenvalue ♯ . If = ∅ then it easily follows from the formula (16) that ∅ = 1. Therefore ℒ ♯ ( ∅ ) = 0 and we arrived to an identity Using this identity it is not difficult to see that So we see from the formula (19) that is an eigenfunction of ℒ ♯ with eigenvalue and it is not difficult to verify that the last expression is ♯ . Theorem is proved. Proof. It is enough to apply Theorem 5 to the family . All the conditions of this Theorem are fulfilled except the condition ∈ , . But this condition easily follows from the Pieri formula and induction on | |.