Newell-Littlewood numbers II: extended Horn inequalities

The Newell-Littlewood numbers $N_{\mu,\nu,\lambda}$ are tensor product multiplicities of Weyl modules for classical Lie groups, in the stable limit. For which triples of partitions $(\mu,\nu,\lambda)$ does $N_{\mu,\nu,\lambda}>0$ hold? The Littlewood-Richardson coefficient case is solved by the Horn inequalities (in work of A. Klyachko and A. Knutson-T. Tao). We extend these celebrated linear inequalities to a much larger family, suggesting a general solution.

Let g be a semisimple complex Lie algebra, Λ + be the set of dominant integral weights, and L g be the root lattice. Suppose V λ is the irreducible representation of g indexed by λ ∈ Λ + . Define multiplicities m λ µ,ν by The tensor semigroup is Tensor(g) = {(µ, ν, λ) ∈ Λ 3 + : m λ µ,ν > 0}. Compare this with the saturated tensor semigroup, where w 0 is the longest length element of the Weyl group associated to g.
There are generalized Horn inequalities describing SatTensor(g) [2]. Since N µ,ν,λ is a tensor product multiplicity for g of classical type B, C, D, these results are related to our classification problem, but do not solve it. Classifying N µ,ν,λ > 0 concerns Tensor(g) rather than the possibly different SatTensor(g). In type A, the saturation theorem [11] implies Tensor(sl(n)) = SatTensor(sl(n)).
For the other classical types, saturation is either false, or not known (see [17,13]). 1 N. Ressayre [16] introduces different generalized Horn inequalities that hold when the Kronecker coefficient g µ,ν,λ is nonzero. Those coefficients are also tensor product multiplicities, but for Specht modules, not Weyl modules.

Main results.
We suggest an answer to our problem, by introducing a large, new family of inequalities extending (2).
In contrast with [2,16], our methods are completely combinatorial, starting from (1). The main work was the uncovering of the form of the inequalities (3).
is the partition obtained by taking the complement of θ ⊆ n × m and rotating 180-degrees.
We use the following reformulation of the main definition.

Lemma 2.2. In Definition 1.2, it is equivalent to replace Condition (III)(3) with
The "saturated version" of this conjecture is now [7, Theorem 1.5]. 3 The proof of this direction is now [7, Corollary 10.5].
Proof. Notice that where for a partition α, we denote α ′ to be the transpose of α.
A similar equality holds for the other two arguments. Hence Since, by Condition (II), |C| = |A 1 | + |B 2 |, we can apply Theorem 2.1 and obtain The other two cases are similarly proved to be equivalent with the corresponding condition in Definition 1.2.

Special subclasses of the inequalities.
In [6] we proved: Theorem 2.4 (Extended Weyl inequalities). Let (µ, ν, λ) ∈ Par 3 n and 1 ≤ k ≤ i < j ≤ l ≤ n, let m = min(i − k, l − j) and M = max(i − k, l − j). If N µ,ν,λ > 0 then Definition 2.5. For disjoint X, Y ⊆ [n], the subset-sum inequalities are and so (v) can be rewritten as This is a subset-sum inequality, and we are done by (iii).
Example 2.7. The extended Horn inequalities for n = 2 are the S 3 -permutations (6) of: where (10) and (11) are Horn inequalities, (12) is a triangle inequality, and (13) is both an extended Weyl inequality and a subset-sum inequality.
Since there are L rows where at least one of µ i or ν i is nonzero, this means that there are L − |µ| − |ν| + 2|µ ∧ ν| rows such that µ i = ν i > 0, so it suffices to prove that Rearranging the terms and substituting in for the definition of k, this becomes Multiplying the above expression by 2 and using the definition of | · |, we get: We split the remainder of the proof of the claim into two cases: whether L < p or L ≥ p.
Case 1: (L < p) Here we can rewrite the above inequality as However, this is always true, since which can be seen by letting The slightly trickier verification needed from (III) (3) is the latter is obvious. The check c τ (A) τ (B 1 ),τ (C 2 ) > 0 is analogous. Case 2: (L ≥ p) Here we can instead rewrite the above inequality as and so it suffices to show This is true since (Y, X, X, Y, [L], ∅) ∈ G n is just a subset-sum inequality.
As a result, one can remove a vertical strip from µ of length |µ|+p−|ν| 2 , and then add a vertical strip of length |ν|+p−|µ| 2 back in to result in ν. This is exactly the conjugate statement of Proposition 2.4 of [6], so N µ,ν,(1 p ) > 0.
summer program. We also thank the anonymous referees for their useful comments. We used Anders Buch's Littlewood-Richardson calculator in our experiments. This work was partially supported by a Simons Collaboration grant, funding from UIUC's Campus Research Board, and an NSF RTG grant. SG was partially supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-1746047.
AY would like to add an additional acknowledgement to Ian Goulden and David Jackson, who were instrumental in said authors' professional development at Waterloo. Your words were heard, and made a difference.