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#Hives and BZ-polytopes

Knutson–Tao hives and Berenstein–Zelevinsky patterns are polyhedral models for Littlewood–Richardson coefficients. They replace tableaux by lattice points in a polytope whose boundary data is given by three partitions. The main references are [BZ88, BZ92, KT99].

For fixed partitions \(\lambda,\mu,\nu,\) the corresponding hive polytope is cut out by linear inequalities on a triangular array. The boundary differences encode \(\lambda,\mu,\nu,\) and each elementary rhombus imposes a concavity inequality.

The three colored rhombi indicate the three local orientations of hive inequalities.

That is, every elementary rhombus in the triangular array, in each of the three orientations shown, must satisfy the same local rule: if \(a,d\) are one opposite pair and \(b,c\) are the other, then \[a+d \geq b+c.\]

With the usual boundary convention, \[c^\nu_{\lambda,\mu} = |\{\text{integer hives with boundary }(\lambda,\mu,\nu)\}|.\] Equivalently, one may use Berenstein–Zelevinsky patterns; these are different coordinates for the same Littlewood–Richardson cone.

Example

The product \[\schurS_{(2)}\schurS_{(1)} = \schurS_{(3)}+\schurS_{(2,1)}\] says that the two relevant hive polytopes each contain one integer point, and all other boundary triples for this product contain none.

#Skew hives

T. Le and S. Nguyen introduce skew hives and skew skeps, whose integer points compute the structure constants in products of two skew Schur functions [LN26]. Ordinary hives and Speyer’s skeps occur as special cases. They prove that if the two pairs of boundary shapes \((\lambda,\mu)\) and \((\nu,\rho)\) are replaced by two points \((\widetilde\lambda,\widetilde\mu)\) and \((\widetilde\nu,\widetilde\rho)\) in their minimal type-\(A\) alcoved parallelepiped, with the same sum, then \[\schurS_{\widetilde\lambda/\widetilde\mu} \schurS_{\widetilde\nu/\widetilde\rho} -\schurS_{\lambda/\mu}\schurS_{\nu/\rho}\] is Schur-positive. This extends Schur log-concavity to skew shapes and gives log-concavity consequences for Newell–Littlewood numbers and shadow skew Schur functions. The same work gives bijections among skew hives, skew skeps, and phased peelable tableaux.

#Saturation

The hive model gives a geometric proof of the saturation theorem: \[c^\nu_{\lambda,\mu}\gt{}0 \quad\Longleftrightarrow\quad c^{k\nu}_{k\lambda,k\mu}\gt{}0 \qquad (k\geq 1).\] This was proved by A. Knutson and T. Tao [KT99]; see also [Buc00]. In hive language, scaling the boundary scales the polytope. Saturation says that rational feasibility already detects integral feasibility.

#Stretching and Ehrhart theory

The stretching function \[k\mapsto c^{k\nu}_{k\lambda,k\mu}\] is naturally an Ehrhart-type lattice-point count. The polynomiality of this function is nontrivial and was proved by H. Derksen and J. Weyman [DW02], and by E. Rassart [Ras04]. The polynomial is called a stretched Littlewood–Richardson polynomial .

Conjecture (King–Tollu–Toumazet, [KTT04]).

The stretched Littlewood–Richardson polynomial \[k\mapsto c^{k\nu}_{k\lambda,k\mu}\] has nonnegative coefficients.

This is one of the motivating examples for the Ehrhart positivity conjectures on the Ehrhart page. The same circle of ideas also contains stretched Kostka coefficients, flagged skew Schur coefficients, and Ehrhart polynomials for key polynomials.

#Related models

Hives are closely related to several other models:

Bibliography

  1. [BZ88]A. D. Berenstein and A. V. Zelevinsky. Tensor product multiplicities and convex polytopes in partition space. Journal of Geometry and Physics, 5(3):453–472, 1988.
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  8. [LN26]Tuong Le and Son Nguyen. Skew Hives, Skew Skeps, Skew Schur Log-Concavity. arXiv:2608.13544v1, 2026.
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