Young Flattenings in Tensors & Polynomials

• Young flattenings are representation-theoretic generalized flattenings of tensors and homogeneous polynomials using Schur functors and Pieri maps.
• They provide lower bounds for border rank and yield determinantal equations for secant varieties, refining classical matrix flattenings.
• Their multiplicative properties and efficient algorithmic implementations make them crucial for tensor decomposition and algebraic complexity studies.

Young flattenings are representation-theoretic generalized flattenings of tensors and homogeneous polynomials, built from Schur functors and Pieri maps, and used to produce determinantal equations for secant varieties together with lower bounds for border rank. In the literature summarized here, they appear both in multilinear form for tensors and in symmetric form for polynomials, where they refine ordinary matricizations by exploiting the full $GL(V)$-module structure rather than only a partition of tensor indices. They are central to several strands of work on tensor rank, border rank, secant geometry, and algebraic complexity, including monomial border ranks, Chow and Veronese secant varieties, determinant and permanent complexity, and recent algorithmic developments (Christandl et al., 2017, Oeding, 2016).

1. Definition and representation-theoretic construction

Ordinary flattenings rearrange tensor indices to form a matrix; for $t \in V_1 \otimes V_2 \otimes V_3$, one may group modes as $V_1 \mid (V_2 \otimes V_3)$, $V_2 \mid (V_1 \otimes V_3)$, or $V_3 \mid (V_1 \otimes V_2)$. Generalized flattenings replace such reshaping by an arbitrary linear map

$$F: V_1 \otimes V_2 \otimes V_3 \to V_1' \otimes V_2',$$

and Young flattenings are a special, representation-theoretic kind of generalized flattening using the Pieri rule and Schur functors (Christandl et al., 2017).

For symmetric tensors, Young flattenings are described in terms of partitions $\lambda,\mu$ and Schur modules $S_\lambda V, S_\mu V$. One construction starts from a $GL(V)$-equivariant Pieri inclusion

$S_\mu V \to S^dV \otimes S_\lambda V,$

and contraction with $t \in V_1 \otimes V_2 \otimes V_3$0 yields a linear map

$t \in V_1 \otimes V_2 \otimes V_3$1

A second standard presentation writes the Pieri map as

$t \in V_1 \otimes V_2 \otimes V_3$2

with $t \in V_1 \otimes V_2 \otimes V_3$3, and the associated flattening as

$t \in V_1 \otimes V_2 \otimes V_3$4

The cited literature explicitly uses these equivalent descriptions in different module conventions (Oeding, 2016, Haas et al., 2021).

In suitable bases of semistandard Young tableaux, Young flattenings are represented by matrices whose entries are homogeneous polynomials in the coefficients of the input polynomial. This tableau-level realization is not merely notational: it is the basis for explicit computation, for rank calculations, and for the derivation of determinantal equations for secant varieties (Oeding, 2016, Haas et al., 2021).

2. Rank certificates and secant-variety equations

The operational significance of a flattening is that matrix rank furnishes a border-rank certificate. For a generalized flattening $t \in V_1 \otimes V_2 \otimes V_3$5, the basic estimate is

$t \in V_1 \otimes V_2 \otimes V_3$6

where the maximum is taken over simple tensors $t \in V_1 \otimes V_2 \otimes V_3$7. This is the framework in which Young flattenings are used as lower-bound methods (Christandl et al., 2017).

For symmetric tensors, the standard border-rank criterion is expressed relative to the reference point $t \in V_1 \otimes V_2 \otimes V_3$8. If

$t \in V_1 \otimes V_2 \otimes V_3$9

and $V_1 \mid (V_2 \otimes V_3)$0 has border rank $V_1 \mid (V_2 \otimes V_3)$1, then $V_1 \mid (V_2 \otimes V_3)$2 has rank at most $V_1 \mid (V_2 \otimes V_3)$3. Hence, if $V_1 \mid (V_2 \otimes V_3)$4 has rank $V_1 \mid (V_2 \otimes V_3)$5, then

$V_1 \mid (V_2 \otimes V_3)$6

This rank-quotient principle is the standard bridge from representation-theoretic linear algebra to border-rank lower bounds (Oeding, 2016).

The same matrices also provide equations for secant varieties. The vanishing of suitable minors of a Young flattening matrix gives determinantal equations for secant varieties of the Veronese and related varieties. In the Chow setting, Koszul Young flattenings produce minors that lie in the ideal of $V_1 \mid (V_2 \otimes V_3)$7, and analogous rank bounds yield set-theoretic equations for secant varieties $V_1 \mid (V_2 \otimes V_3)$8 (Guan, 2015).

3. Tensor products and multiplicativity phenomena

A notable structural property is that lower bounds on border rank obtained from generalized flattenings, including Young flattenings, multiply under tensor product. If $V_1 \mid (V_2 \otimes V_3)$9 are tensors and $V_2 \mid (V_1 \otimes V_3)$0 are generalized flattenings for them, then for $V_2 \mid (V_1 \otimes V_3)$1,

$V_2 \mid (V_1 \otimes V_3)$2

The same specialization applies to Young flattenings built from Pieri inclusions (Christandl et al., 2017).

The linear-algebraic reason is that the tensor-product flattening is constructed as $V_2 \mid (V_1 \otimes V_3)$3, and matrix rank is multiplicative under Kronecker product. The representation-theoretic reason given in the cited work is that the Pieri inclusions and their ranks on simple tensors remain compatible under tensor product, so the denominator retains the same factorized form (Christandl et al., 2017).

This multiplicativity sharply contrasts with tensor rank itself. Tensor rank is submultiplicative under tensor product, but not multiplicative in general: if a tensor $V_2 \mid (V_1 \otimes V_3)$4 has border rank strictly smaller than rank, then the tensor rank of a sufficiently high tensor power $V_2 \mid (V_1 \otimes V_3)$5 is strictly smaller than $V_2 \mid (V_1 \otimes V_3)$6. By contrast, the flattening-derived lower bounds multiply exactly, even though multiplicativity of border rank and asymptotic rank remains open (Christandl et al., 2017).

4. Monomials, optimal shapes, and Chow-type applications

For monomials, the effectiveness of Young flattenings depends strongly on the choice of partition shape. For a monomial

$V_2 \mid (V_1 \otimes V_3)$7

a monomial-optimal shape is defined by

$V_2 \mid (V_1 \otimes V_3)$8

Under the dominance conditions stated in the monomial study,

$V_2 \mid (V_1 \otimes V_3)$9

This yields the best possible lower bound for large classes of monomials, including all monomials up to degree $V_3 \mid (V_1 \otimes V_2)$0, monomials in $V_3 \mid (V_1 \otimes V_2)$1 variables, and any power of the product of variables (Oeding, 2016).

The same work also delineates the limits of the method. For degree $V_3 \mid (V_1 \otimes V_2)$2 and higher there are monomials for which no Young flattening can give a lower bound that matches the conjecturally tight upper bound of Landsberg and Teitler. The paper therefore presents Young flattenings not as universally optimal, but as a powerful family with a describable ceiling on monomial problems (Oeding, 2016).

Koszul Young flattenings sharpen this picture for products of variables and Chow varieties. For $V_3 \mid (V_1 \otimes V_2)$3, $V_3 \mid (V_1 \otimes V_2)$4, and $V_3 \mid (V_1 \otimes V_2)$5, the Koszul Young flattening is

$V_3 \mid (V_1 \otimes V_2)$6

For $V_3 \mid (V_1 \otimes V_2)$7, the paper computes the rank explicitly as $V_3 \mid (V_1 \otimes V_2)$8, proves that the minors of size $V_3 \mid (V_1 \otimes V_2)$9 lie in the ideal of the Chow variety $$F: V_1 \otimes V_2 \otimes V_3 \to V_1' \otimes V_2',$$0, and derives the lower bound

$$F: V_1 \otimes V_2 \otimes V_3 \to V_1' \otimes V_2',$$1

which improves over the classical flattening bound $$F: V_1 \otimes V_2 \otimes V_3 \to V_1' \otimes V_2',$$2 for odd $$F: V_1 \otimes V_2 \otimes V_3 \to V_1' \otimes V_2',$$3 (Guan, 2015).

5. Determinant, permanent, and invariant-tensor applications

Young flattenings and their Koszul specializations are deeply connected to algebraic complexity, especially for determinant and permanent polynomials. In the symmetric setting, Koszul-Young flattenings were used to obtain new lower bounds for the symmetric border rank of the $$F: V_1 \otimes V_2 \otimes V_3 \to V_1' \otimes V_2',$$4 determinant for all $$F: V_1 \otimes V_2 \otimes V_3 \to V_1' \otimes V_2',$$5, as well as further lower bounds for the $$F: V_1 \otimes V_2 \otimes V_3 \to V_1' \otimes V_2',$$6 permanent. For $$F: V_1 \otimes V_2 \otimes V_3 \to V_1' \otimes V_2',$$7, the resulting bounds improve the earlier $$F: V_1 \otimes V_2 \otimes V_3 \to V_1' \otimes V_2',$$8 estimate by explicit rational correction factors; for $$F: V_1 \otimes V_2 \otimes V_3 \to V_1' \otimes V_2',$$9, the paper proves $\lambda,\mu$0, and for $\lambda,\mu$1, it gives

$\lambda,\mu$2

The improvement is in the lower-order term rather than the leading exponential term (Farnsworth, 2015).

For ordinary tensor rank, recursive Koszul flattenings extend the same representation-theoretic philosophy to determinant and permanent tensors. In that framework, Koszul flattenings are described as Schur functors attached to hook shapes $\lambda,\mu$3, and recursive application across different tensor legs yields stronger lower bounds. The paper proves exact tensor ranks

$\lambda,\mu$4

over arbitrary field of characteristic $\lambda,\mu$5, and obtains lower bounds on $\lambda,\mu$6 that completely separate determinant and permanent tensors by tensor rank growth (Han et al., 15 Mar 2025).

A further development concerns $\lambda,\mu$7-invariant tensors in $\lambda,\mu$8 arising from matrices of constant rank. This work gives the first explicit use of Young flattenings for tensors beyond Koszul to obtain border rank lower bounds. In the example

$\lambda,\mu$9

with $S_\lambda V, S_\mu V$0 and $S_\lambda V, S_\mu V$1, the first Koszul flattening gives $S_\lambda V, S_\mu V$2, while the Young flattening gives $S_\lambda V, S_\mu V$3, and ten is maximal. The paper also determines the border rank of three tensors and emphasizes that general Young flattenings can be strictly stronger than Koszul flattenings, especially for unbalanced tensors (Wu, 2024).

6. Bases, implementations, algorithmic uses, and newer extensions

The computational realization of Young flattenings depends heavily on the chosen model of irreducible polynomial representations. One line of work revisits Young flattenings in the Schur module basis, clarifying that the box-filling procedure described by Oeding and Farnsworth is not equivariant, and deriving explicit equivariant Pieri inclusions in the Schur basis. Using Reuven Hodges’ tableau straightening algorithm as a subroutine, the resulting implementation outperforms Steven Sam’s PieriMaps implementation by several orders of magnitude on many examples, in particular for powers of linear forms, where the reported speedup is over a factor of $S_\lambda V, S_\mu V$4 (Haas et al., 2021).

Koszul–Young flattenings have also acquired an algorithmic role in tensor decomposition. For an $S_\lambda V, S_\mu V$5 tensor with $S_\lambda V, S_\mu V$6, $S_\lambda V, S_\mu V$7, and $S_\lambda V, S_\mu V$8, an algorithm based on these flattenings is guaranteed to succeed for generic components when

$S_\lambda V, S_\mu V$9

for arbitrary $GL(V)$0, with runtime polynomial in $GL(V)$1. In the square case $GL(V)$2, this is a factor-of-2 improvement over classical simultaneous diagonalization, which requires $GL(V)$3. The same paper proves a barrier result: no flattening of the style considered can surpass rank $GL(V)$4, and for $GL(V)$5 tensors, degree-$GL(V)$6 polynomial flattenings cannot surpass rank $GL(V)$7 for a constant $GL(V)$8 (Kothari et al., 2024).

Recent nonlinear work places classical Young flattenings inside a wider family. Kronecker-Koszul flattenings generalize Koszul flattenings and related secant-variety equations, while Kronecker-Young flattenings replace wedges by general Schur functors on grouped tensor powers. In this setting, tangency flattenings provide the first explicit polynomial equations vanishing on secant varieties of the Segre variety but not on cactus varieties. This suggests that the classical linear Young-flattening paradigm is now part of a broader determinantal toolkit in which representation-theoretic flattenings remain the foundational special cases (Doležálek et al., 13 Feb 2026).