Kumar–Torres Branching Model

• The Kumar–Torres branching model is a combinatorial description that computes branching coefficients from GL₂n(ℂ) to Sp₂n(ℂ) using flagged hives and rhombus inequalities.
• It establishes canonical bijections between flagged hives, Sundaram tableaux, and Kwon tableaux, linking them via crystal-commuting operators.
• The model resolves the Lecouvey–Lenart conjecture by unifying tableau and crystal models, thereby advancing symplectic representation theory.

The Kumar–Torres branching model provides a combinatorial description of the branching coefficients that arise in the restriction of irreducible representations from the general linear group $\mathrm{GL}_{2n}(\mathbb{C})$ to the symplectic group $\mathrm{Sp}_{2n}(\mathbb{C})$. The central objects in this model are flagged hives, which are discrete structures subject to rhombus inequalities, boundary conditions, and additional flag constraints. The model forms a bridge between diverse tableau and crystal models used in the literature, canonically linking the Sundaram and Kwon branching paradigms via explicit combinatorial bijections and crystal-commuting operators. This approach has resolved outstanding conjectures on bijections between these models, notably the Lecouvey–Lenart conjecture, and has deepened the understanding of the interplay between combinatorics and symplectic representation theory (Azenhas, 11 Jan 2026).

1. Definition and Construction of the Kumar–Torres Branching Model

Let $n \ge 1$ and fix partitions $\lambda$, $\mu$, and $\nu$ such that $\ell(\lambda) \le 2n$, $\ell(\mu) \le n$, $\ell(\nu) \le 2n$, with $\nu$ even (all column lengths even). The model addresses the branching coefficient

$$c_\mu^\lambda = \dim\,\mathrm{Hom}_{\mathrm{Sp}_{2n}}\left(V^{\mathrm{GL}_{2n}}(\lambda),\,V^{\mathrm{Sp}_{2n}}(\mu)\right)$$

by counting the number of $(\lambda,\mu,\nu)$-flagged hives of size $2n$. A hive is a function

$$H : \Delta_{2n} \to \mathbb{Z}_{\ge 0}, \qquad \Delta_{2n} = \{(i,j,k) \in \mathbb{Z}_{\ge 0}^3 : i+j+k=2n\}$$

satisfying rhombus inequalities for each rhombus in the triangular array:

• $$H(i,j,k) + H(i+1,j-1,k+1) \ge H(i+1,j-1,k) + H(i,j-1,k+1)$$,
• and two cyclically rotated versions.

Boundary values are enforced by setting:

• $H(i,2n-i,0) = \sum_{r=1}^i \lambda_r$,
• $H(i,0,2n-i) = \sum_{r=1}^i \mu_r$,
• $H(0,i,2n-i) = \sum_{r=1}^i \nu_r$, for $i=0,\dots,2n$.

The model incorporates flag conditions, prescribing for $r=0,1,\dots,n$:

• $H(r, n+r, n-r) \le \sum_{s=1}^r \mu_s + r$,
• $H(n-r, n+r, r) \ge \sum_{s=1}^r \mu_s - r$.

The flagged hive model was introduced by Kushwaha, Raghavan, and Viswanath [23,24], and is equivalent to their original flag conditions.

2. Properties of Flagged Hives

A flagged hive must satisfy all rhombus inequalities and boundary conditions described above. These requirements define a bounded region in the lattice of integral hives, and the flag conditions impose additional linear constraints along the “middle” edges of the hive triangle. The flagged hive model recovers and generalizes the structure of the classical hive polytope, encoding not only the combinatorial data of weights $\lambda$, $\mu$, and $\nu$ but also the symplectic features relevant for the restriction to $\mathrm{Sp}_{2n}$.

The constraints can be summarized as follows:

Condition	Type	Explicit Formulation
Rhombus inequalities	Combinatorial (polyhedral)	See inequalities (a)–(c) in the main text above
Boundary conditions	Partition sums	$H(i,2n-i,0) = \Lambda_i$, etc.
Flagged-hive conditions	Linear flag constraints	$H(r,n+r,n-r)\leq M_r+r$, $H(n-r,n+r,r)\geq M_r-r$ for $r=0,\dots,n$

These conditions generate the set whose cardinality yields $c_\mu^\lambda$, with an implicit union over all possible even partitions $\nu$ (Azenhas, 11 Jan 2026).

3. Crystal Structures and Tableau Bijections

The model canonically relates three central combinatorial objects:

• Flagged hives with boundary $(\lambda,\mu,\nu)$,
• Littlewood–Richardson–Sundaram (LRS) tableaux of shape $\lambda/\mu$ and weight $\nu$,
• Sets of LR tableaux $T\in LR(\lambda/\nu,\mu)$ such that $\mathrm{Evac}(T)$ satisfies Kwon’s symplectic (King semistandard) condition.

The main theorem asserts natural bijections:

$\{\text{flagged hives with boundary }(\lambda,\mu,\nu)\} \longleftrightarrow LRS(\lambda/\mu,\nu) \longleftrightarrow \{T\in LR(\lambda/\nu,\mu)\colon\text{$\mathrm{Evac}(T)$ is symplectic}\}$

Kushwaha, Raghavan, and Viswanath [23,24] established a crystal-commuter $R$ acting on flagged hives, inducing a $GL_{2n}$-crystal structure whose isotypical components match the LRS tableaux. Henriques and Kamnitzer’s $gl_n$ crystal-commuter identifies the flagged-hive model with skew LR tableaux under Sundaram’s rule (odd letters below row $n+i$ are forbidden).

Lecouvey and Lenart provided a spinor-model rule for $(GL_{2n}, Sp_{2n})$ via right companions and Schützenberger evacuation. Kumar and Torres showed the explicit path connecting these structures in the hive polytope, confirming the canonical equivalence of the flagged hive, Sundaram tableau, and Kwon tableau models (Azenhas, 11 Jan 2026).

4. Companion Tableaux and Evacuation: The Sundaram–Kwon Bijection

Let $T\in LR(\lambda/\mu,\nu)$. The right companion $\mathsf{RC}(T)$ and left companion $\mathsf{LC}(T)$ are semistandard tableaux extracted from $T$ via Gelfand–Tsetlin patterns corresponding to different subsets of the shape. Explicitly:

• The right companion $\mathsf{RC}(T)$ is the tableau recording where entries $\{1,\dots,r\}$ sit in the skew shape $\lambda/\nu$.
• The left companion $\mathsf{LC}(T)$ records the complementary nested shapes for $\{r+1,\dots,2n\}$.

Azenhas [A18] demonstrated (Theorem 4.10) that

$\mathsf{RC}(T) = \mathrm{Evac}(\mathsf{LC}(T)),$

and further, the left companion of a Sundaram tableau always satisfies the King–semistandard condition:

$$\mathsf{LC}(S)(i,1) \ge 2i-1,\quad \text{for all } i=1,\dots,\ell(\mu).$$

This bijective correspondence underpins the connection between Sundaram and Kwon tableaux: the image of LRS tableaux under left-companion construction are precisely those tableaux whose evacuation defines a symplectic tableau (King condition).

An explicit algorithm for this bijection involves reading and recording Gelfand–Tsetlin patterns from the tableau $S$, then verifying the King–semistandard condition entrywise (Azenhas, 11 Jan 2026).

5. Explicit Examples

Worked examples illustrate the above constructions and the role of the King–semistandard condition.

Example 1: For $n=4$, $\lambda=(6,5,5,4,3,1)$, $\mu=(4,3,2,1)$, $\nu=(4,4,3,3)$, consider the Sundaram tableau

$S = \young(\,\,,\,\,1,1,\,\,;\,\,,1,2,\,\,;\,\,,1,2,3;\,;\,,2,3,4;\,;\,,3,4)$

of shape $\lambda/\mu$ and weight $\nu$. Its left companion $G=\mathsf{LC}(S)$ is:

$G = \young(3\,4\,6\,\,,\,4\,5\,7,\,6\,8,\,8)$

(first column $(8,6,4,3)$ bottom to top). The King–semistandard condition is violated ($G(4,1)=3 < 7$), so $G$ is not symplectic.

Example 2: With another tableau $S'$ avoiding forbidden odd entries, the left companion $G'=\mathsf{LC}(S')$ has first column $(8,7,5,4)$, and all King–semistandard inequalities are satisfied, so $G' \in SpT_{2n}(\mu)$, demonstrating the bijection.

6. Significance and Resolution of the Lecouvey–Lenart Conjecture

The Kumar–Torres model, particularly through its flagged hive interpretation, has settled the Lecouvey–Lenart conjecture by providing explicit combinatorial bijections between the branching models due to Kwon and Sundaram for the pair $(GL_{2n}(\mathbb{C}), Sp_{2n}(\mathbb{C}))$. The canonical isomorphism between the crystal structures validates the robustness and universality of the flagged hive approach (Azenhas, 11 Jan 2026).

The construction elucidates the geometric and polyhedral underpinnings of branching for the symplectic pair, clarifying the tableau-theoretic and crystal-commuter correspondences central to modern algebraic combinatorics and representation theory.