Superbinomial coefficients

We investigate several families of polynomials that are related to certain Euler type summation operators. Being integer valued at integral points, they satisfy combinatorial properties and nearby symmetries, due to triangle recursion relations involving squares of polynomials. Some of these interpolate the Delannoy numbers. The results are motivated by and strongly related to our study of irreducible Lie supermodules, where dimension polynomials in many cases show similar features.


Introduction
We are interested in a family of polynomials p(n, x) that, for particular polynomials a(n, x), satisfy recursion formulas of the form p(n, x + 1) + 2 p(n, x) + p(n, x − 1) = a(n, x) 2 .
Evaluated at natural numbers x = m, these polynomials define integers p(n, m) with interesting combinatorial properties and representation theoretic interpretations.
To illustrate this, let us consider the Pochhammer polynomials. Their values at integral points give the binomial coefficients for the classical Pascal triangle, which, among others, can be considered as dimensions of certain irreducible representations of the special linear group SL(n). The isomorphism classes of finite dimensional irreducible representations of the special linear group SL(n) over C are described by their dominant weights λ, parameterized by integers λ 1 ≥ λ 2 ≥ · · · ≥ λ n−1 ≥ 0. The symmetric powers S m (C n ) of the n-dimensional standard representation C n of SL(n) are the irreducible representations of dimension m+n−1 m−1 that are obtained for λ 1 = m and λ i = 0, i ≥ 2. By an index shift, restoring the symmetry between n and m, for n, m ≥ 1 we define P cl (n, m) := dim(S m−1 (C n )) = (m − 1) + (n − 1) n − 1 .
The numbers P cl (n, m) are values of rational polynomials p cl (n, x) of degree n − 1 in the variable x. Together with the initial condition p cl (0, x) = 0, they satisfy p cl (n, m) = p cl (m, n) + δ n1 · δ m0 for all integers n, m ≥ 0. This almost symmetry condition and the initial condition uniquely characterize the polynomials p cl (n, x), such that p cl (1, x) = 1 and p cl (n, x) = (x + n − 2) · · · x/(n − 1)! holds for all n ≥ 2. Hence, p cl (n, m) coincides with dim(S m−1 (C n )) = m+n−2 n−1 for m, n ≥ 1. If we formally set dim(S −1 (C n )) = 0, the dimension is given by p cl (n, m) for all n, m ≥ 0, except for the case (n, m) = (1, 0).
In analogy, here we consider polynomials p(n, x) of degree ≤ 2(n − 1) with p(0, x) = 0, such that the values p(n, m)+(−1) m+n ·n are symmetric for all n, m ≥ 0. Imposing the additional conditions p(n, x) = p(n, −x), this uniquely determines the polynomials p(n, x). For m ≥ n, we further define P(n, m) = p(n, x)| x=m . Extended by symmetry P(m, n) := P(n, m), these numbers (up to a simultaneous index shift of n, m by 1) will be called the superbinomial coefficients. We refer to Tables 1 and 2 for the first values of p(n, m) and P(n, m), respectively.
In order to relate these superbinomial coefficients P(n, m) to the classical binomial coefficients, let us explain how they arise from the representation theory of the superlinear groups SL(n|n). The isomorphism classes of finite dimensional irreducible representations are again described by dominant weights λ [7,9]. The superdimensions of these representations are zero unless λ is maximal atypical [6,10,13]. The maximal atypical λ are again parameterized by integers λ 1 ≥ λ 2 ≥ · · · λ n−1 ≥ 0. For λ 1 = m ≥ 0 and λ i = 0 for i ≥ 2 as before, let S m denote the corresponding irreducible maximal atypical representations of SL(n|n). Notice, the representations S m are no longer isomorphic to the symmetric powers of some representations of SL(n|n). However, for m ≥ 1 we have for the dimension of the irreducible representation S m−1 of SL(n|n) Observe the index shift, in analogy with the dimensions P cl (n, m) for the classical case. Next, in the relation between p(n, m) and P(n, m), the condition m ≥ n now really becomes significant. Indeed, for fixed n the dimensions do not depend on m in a polynomial way in the range m < n, as we briefly discuss eventually. Even more interestingly, the polynomials p(n, x) interpolate the superdimension p(n, 0) = sdim(S m−1 ) at the point x = 0 < n, where p(n, x) a priori does not have an obvious dimensional interpretation. This holds true in similar cases for more general SL(n|n) modules; see [8]. For character and dimension formulas for irreducible Lie supermodules of SL(n|n) in general, we refer to [2,4,9,12].
Returning to our main interest, in this note we focus on particular polynomial relations satisfied by the polynomials p(n, m). The first type of relations is for n ∈ N 0 and m ∈ N, where A : It is not hard to see that, for fixed n, there exist polynomials a 2 (n, x) of degree 2(n −1) such that A(n, m) = a 2 (n, x)| x=m holds for all integers m ≥ 1. Let us define an Euler type summation operator . Since this operator acts bijectively on the polynomial ring Q[x], the above relations characterize the polynomials f (x) = p(n, x) as the unique solutions of the polynomial equations In Definition 3.1, we similarly give polynomials a 1 (m, x) satisfying a 1 (m, x)| x=n = A(m, n). In Theorem 3.2, we show the first as well as second variable summation equations respectively, As polynomials identities, these equations hold for all x. In particular, p(n, x) is of degree 2(n − 1) in x. This implies the polynomial identity na 1 (n, x) = xa 2 (n, x) (Proposition 3.5), which amounts to the symmetry n A(m, n) = m A(n, m). Besides the summation equations above, there also exists a summation relation of second kind Here, d(n, x) turn out to be the Delannoy polynomials, defined in Sect. 4. However, we do not know any representation theoretic interpretations of the results in Sect. 4. This comes from the fact that such a result would relate representations of the groups SL(n|n) for different n.
Although the connection of the polynomials p(n, x) and their combinatorial properties to representation theory is not the primary focus of this paper, our interest arose while searching for alternatives to the combinatorial character formulas [2,9], in order to obtain and understand dimension formulas. In fact, for atypical irreducible representations [7] these character formulas are not entirely satisfying, because the known formulas turn out to be "rather intricate and difficult to apply" [4]. We therefore looked for a different approach to dimension formulas: In [8], we conjectured that for all irreducible maximal atypical representations of weight λ that are attached to a fixed basic weight λ basic (see [6]), the dimensions of the representations depend on the coefficients of the weight λ in a polynomial way. Notice that, for fixed n, the number of different basic weights λ basic is finite and given by the Catalan number C n . Hence, this conjecture predicts the existence of C n (usually different) dimension polynomials for fixed n. For example, the highest weights of the representations S k are basic in the sense above if and only if 0 ≤ k ≤ n − 1. For their basic weights λ basic , consider the set of isomorphy classes of the irreducible maximal atypical representations of weight λ attached to λ basic . These sets are singletons for k < n − 1, represented by S k . For k = n − 1, on the other hand, this set is infinite and its representatives are the representations S m for m ≥ k. For k < n − 1, the associated dimension polynomials are constant and equal to P(n, k). However, for k = n − 1, the dimension polynomial is our polynomial p(n, x) above. Hence, our results on the superbinomial coefficients suggest subtle relations between the C n different dimension polynomials. Furthermore, it could be that the Euler type summation equations, discussed here, are special cases of dimension formulas for certain indecomposable, but not necessarily irreducible modules. A special type of such indecomposable modules are the Kac modules. It is well known that the knowledge of Jordan Hölder constituents of Kac modules is strongly related to the character and dimension formulas. In [8], we also work with other types of indecomposable modules not directly related to Kac modules. We therefore suggest not to ignore other indecomposable modules that are different from Kac modules.

Proposition 2.1 There is a unique family of polynomials
Proof of Proposition 2. 1 We show that the properties (i)-(iv) uniquely define the polynomials p(n, x) by recursion. For n = 0, the polynomial p(0, x) = 0 is fixed by property (i). For n = 1, by (ii) we know p(1, x) = c is a constant polynomial. By (iv), Assuming p(k, x) to be constructed for 0 ≤ k ≤ n, we obtain by property (iv) the following values of p(n + 1, x) Using (iii), we find p(n + 1, −k) = p(n + 1, k). We thus have fixed the values p(n + 1, x) at the 2n + 1 places x ∈ {−n, . . . , 0, . . . , n}. But by (ii), the degree of p(n + 1, x) is at most 2n. Hence, p(n + 1, x) is the unique interpolation polynomial of degree 2n for the values above.
For example, condition (iv) together with (i) implies In particular, The proof of Proposition 2.1 shows that, for all n ≥ 0, the values p(n, k) for k = −n, . . . , n are integers. So by the almost symmetry (iv), for every integer j > 0, the value Let n be a natural number. For integers 0 ≤ μ ≤ n − 1, put for n > 0 satisfy the properties of Proposition 2.1.

Proof of Proposition 2.3 By definition, condition (i) of Proposition 2.1 is satisfied.
For the summands of p(n, x), we have, for all ν, μ, So the same holds true for p(n, x). This implies property (ii). Obviously, In order to prove (iv), which is trivial for n = m, we assume m > n. Substituting Notice that, for m ≤ μ * , the value Table 1 Values p(n, m) of the polynomials p . Hence, condition (iv) of Proposition 2.1 holds for all integers m, n > 0.

Definition 2.4
Let m ∈ N. For integers α and β, we define the natural numbers Hence, for fixed m ∈ N, the function p(n, m) is almost a polynomial of degree 2(m−1) in the first variable n.
iii) For integers m > 0, there is a convenient presentation of t(ν, μ, n; m), In particular, for integers m ≥ n > 0, we obtain Equivalently, using Definition 2.4, for integers m ≥ n > 0,  Hence, the numbers P(n, m) are symmetric.

Summation operators
Consider the Eulerian summation operator E acting on functions f by On monomials, E acts by E(x n ) = n shows that E is bijective on polynomial rings over fields of characteristic = 2. Furthermore, the summation operator E 2 is also well defined on functions f : Z → C.
We have For integers k, consider the following polynomials of degree k with x 0 = 1 and For all integers n ≥ 0, their values coincide with the binomial coefficients n k = n k .

Let p(n, x) be the family of polynomials of Proposition 2.1 defining the numbers p(n, m). Then, for all integers n > 0,
holds. Furthermore, for all integers m > 0, we have Before we prove Theorem 3.2, we deduce some consequences. Consider the symmetric numbers P(n, m) defined in Remark 2.5(iii), which can be written as For m > n, they satisfy P(n, m + i) = p(n, m + i) for i = −1, 0, 1. Similarly, by Proposition 2.1(iv), for m < n, they satisfy P(n, m + i) = p(n, m + i) + (−1) m+i+n (n − m − i) for i = −1, 0, 1. And for m = n, we obtain P(n, n + i) = p(n, n + i) for i = 0, 1, and P(n, n + 1) = p(n, n + 1) − 1. By Theorem 3.2, this implies the following corollary.

Proof of Theorem 3.4 We use the indecomposable SL(n|n)-module
whose dimension is the product of the dimensions of its tensor factors. This immediately implies dim(A S m ) = A(n, m) 2 . By [5,Lemma 4.1], for all m ≥ 1 we have the following decomposition in the Grothendieck ring of representations for m = n, and for m = n we have A S n ∼ S n−2 + S n + 2 · S n−1 + 1. We now list some of the polynomials a 1 (n, x) and a 2 (m, x): The polynomials a 1 (n, x) and a 2 (m, x) satisfy the following properties.

Proposition 3.5 (a)
For all n > 0, there is an identity of polynomials n · a 1 (n, x) = x · a 2 (n, x).

Proof of Proposition 3.5
The ν-th summand of the sum defining a 1 (n, −x) is This equals which up to the factor (−1) n x n is the ν-th summand of a 2 (n, x). This implies (−1) n n · a 1 (n, −x) = x · a 2 (n, x). Property (iii) of Proposition 2.1, i.e. p(n, x) = p(n, −x), is inherited by the images E 2 1 p(n, x) = a 1 (n, x) 2 under the summation operator E 2 1 . So a 1 (n, x) is either even or odd, depending on whether its degree is even or odd. Part (a) follows. Accordingly, a 2 (n, −x) = (−1) n−1 a 2 (n, x) is even or odd. The leading term of the polynomial x m−1 is 1 (m−1)! x n−1 . So the leading term of a 2 (m, x) is The evaluation of a 2 (m, x) at x = 0 only has constituents from ν = 0 and m, so we obtain a 2 (m, 0) = 1 + (−1) m−1 . Similarly, at x = 1 only the terms for ν = 0 and 1 are nonzero, and we obtain a 2 (m, 1) = 2m.
In the rest of this section, we prove Theorem 3.2. We start with a lemma.

Proof of Lemma 3.6
Using the definition of D m (α + 1, β) (see 2.4), the lemma follows by straight forward calculations, distinguishing the cases ν, μ equal to 0, n, or generic. We exemplify this in the generic case 0 < ν, μ < n. The sum b(ν, μ, n, m) here is Summing the first and the second lines as well as the third and fourth, we obtain which simplifies to a(ν, μ, n, m).
Proof of Theorem 3. 2 We show (6) at the integer points x = m > n. Then, both, E 2 1 p(n, x) and a 1 (n, x) 2 being polynomials, coincide. Let m > n be an integer. By changing the summation index, we obtain Hence, a 1 (n, m) 2 = n ν,μ=0 a(ν, μ, n, m) for the numbers (11). On the other hand, using the notation of Remark 2.5(iii), Index shifts ν → ν + 1, μ → μ + 1 in the last sum and in one of the second sums yield Using this, equation (6) now follows from Lemma 3.6. The identity a 1 (n, m) = a 2 (m, n) for integers m, n > 0 becomes obvious after substituting ν → n − ν in a 1 (n, m), noticing that in both sums the summands are actually zero for ν > min(m, n).
Since a 2 (n, m) = A(n, m) by (2), formula (4) follows from (6). Then, equation (7) holds for all integers x = n > 0 and also for all m > 0 due to the symmetry relation satisfied by p(n, m). Exchanging m and n, we obtain (5). Therefore, both being polynomials of degree 2(m − 1) in x, a 2 (m, x) 2 and E 2 2 p(m, x) coincide.  (c) Notice that, by definition, Because, since in this case b(n, 1 + x) = −b(n, x), and by considering the leading term, we obtain b(n, x) = 0 and hence (c). Equation (12) is obvious for n = 0. For n ≥ 1, we compute Using part (a) for 1 + x, respectively, −x, instead of x, the right-hand side of (13) becomes In here, the last term is zero by induction hypothesis. So (12) is proved in general.
The proof of Proposition 4.2(c) essentially amounts to extending the Delannoy array (Table 4)

Proof of Proposition 4.3
It is easy to see that, for n ∈ N, there is a unique family of polynomials q(n, x) in Q[x] satisfying the following properties: (i) For all n, the degree of the polynomial is deg x q(n, x) = 2(n − 1).

Central values
Using the summation operator E f (x) = f (x + 1 2 ) + f (x − 1 2 ), Proposition 4.3 can be reformulated as This suggests that our families of polynomials have interesting properties at halfintegral places. We illustrate this by determining their values at x = 1 2 . By the set of initial values of Proposition 5.1, and by the tribonacci identities satisfied by a 2 (n, x) and d(n, x), we obtain recursion formulas for their values at x = 2k+1 2 . Identity (7) then gives a recursion for the values p(n, 2k+1 2 ). On the other hand, from Comparing coefficients, the first identity of part (a) is proved. For the second one, recall na 1 (n, x) = xa 2 (n, x) from Proposition 3.5.