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Strict unimodality of q-binomial coefficients

We prove strict unimodality of the q-binomial coefficients \binom{n}{k}_q as polynomials in q. The proof is based on the combinatorics of certain Young tableaux and the semigroup property of Kronecker coefficients of S_n representations.

This celebrated result was first conjectured by Cayley in 1856, and proved by Sylvester using Invariant Theory, in a pioneer 1878 paper [17]. In the past decades, a number of new proofs and generalizations were discovered both by algebraic and combinatorial tools, see Section 3. In the previous paper [12], we found a new proof of Sylvester's theorem using combinatorics of Kronecker and Littlewood-Richardson coefficients. Here we use the recently established semigroup property of Kronecker coefficients to prove strict unimodality of q-binomial coefficients: Theorem 1. For all ℓ, m ≥ 8, we have the following strict inequalities: These and the remaining cases are covered in Theorem 6. Note that neither combinatorial nor algebraic tools imply (•) directly, as Sylvester's theorem is notoriously hard to prove and extend. In our previous paper [12], we proved strict unimodality of the diagonal coefficients 2 m m q by combining technical algebraic tools from [13] and Almkvist's analytic unimodality results.
The following result lies in the heart of the proof of the theorem.
Although stated combinatorially, the only proof we know is algebraic. We first establish the lemma and then combine it with computational results to derive the theorem. We conclude with historical remarks, brief overview of the literature and open problems.

Kronecker coefficients
We adopt the standard notation in combinatorics of partitions and representation theory of S n (see e.g. [8,16]). We use g(λ, µ, ν) to denote the Kronecker coefficients: The following technical result was never stated before, but is implicit in [12] (see also [18, §4]).
Proof. Let λ ⊢ n, π ⊢ k and θ ⊢ n − k, and let " * " denote the Kronecker product of symmetric functions. Littlewood's formula states that where c λ µν denote the Littlewood-Richardson coefficients. Clearly, s ν * s a = s ν , for all ν ⊢ a. We obtain: By the Jacobi-Trudi formula, we have: This gives: Taking the coefficient at s µ in the expansion of s λ * s τ k in terms of Schur functions, we get: Substituting this into ( ), gives the result.

Final remarks
3.1. Let us quote a passage from [17] describing how Sylvester viewed his work: "I am about to demonstrate a theorem which has been waiting proof for the last quarter of a century and upwards. [...] I accomplished with scarcely an effort a task which I had believed lay outside the range of human power." The grandeur notwithstanding, it does reveal Sylvester's excitement over his discovery.

3.2.
Proving unimodality is often difficult and involves a remarkable diversity of applicable tools, ranging from analytic to bijective, from topological to algebraic, and from Lie theory to probability. We refer to [1,2,15] for a broad overview of the subject.
3.3. The Additivity Lemma gives an example of a 2-dim Klarner system, which always have a finite basis (see [14]).
3.4. The equation (KOH) in [20], based on O'Hara's combinatorial approach to unimodality of q-binomial coefficients [11], gives a useful recurrence relation (cf. [4,7]). It would be interesting to see if (KOH) can be used to prove Theorem 1. s λ (1, q, . . . , q m ) as a polynomial in q, see [8, p. 137]. We conjecture that if the Durfee square size of λ is large enough, then these coefficients are strictly unimodal. An analogue of (KOH) in this case is in [4].

An important generalization of Sylvester's theorem is the unimodality of
3.6. In a different direction, we believe that for every d ≥ 1 there exists L(d), s.t. p k (ℓ, m) − p k−1 (ℓ, m) ≥ d for all L(d) < k ≤ ℓm/2, and m, ℓ large enough. Unfortunately, the tools in this paper are not directly applicable. However, for ℓ = m, this follows from Prop. 11 in [15], and further extension of Thm. 5.2 in [12] on strict unimodality of the number of partitions into distinct odd parts. Then, combined with Manivel's extension (see Remark 5), and the finite basis theorem (see §3.3), this would prove the conjecture in a similar manner as the proof of Theorem 6. We plan to return to this problem in the future.