2.1 Definitions/Notation

Throughout, $A,B,C,U,V,W$ will denote complex vector spaces, respectively, of dimensions $\mathbf {a},\mathbf {b},\mathbf {c},\mathbf {u},\mathbf {v},\mathbf {w}$ . The dual space to A is denoted $A^*$ . The space of symmetric degree d tensors is denoted $S^dA$ , which may also be viewed as the space of degree d homogeneous polynomials on $A^*$ . Set $\operatorname {Sym}(A):=\bigoplus _d S^dA$ . The identity map is denoted $\operatorname {Id}_A\in A{\mathord { \otimes } } A^*$ . For $X\subset A$ , $X^{\perp }:=\{\alpha \in A^*\mid \alpha (x)=0, \forall x\in X\}$ is its annihilator, and $\langle X\rangle \subset A$ denotes the linear span of X. Projective space is $\mathbb P A= (A\backslash \{0\})/\mathbb C^*$ , and if $x\in A\backslash \{0\}$ , we let $[x]\in \mathbb P A$ denote the associated point in projective space (the line through x). The general linear group of invertible linear maps $A{\mathord {\;\rightarrow \;}} A$ is denoted $\operatorname {GL}(A)$ and the special linear group of determinant one linear maps is denoted $\operatorname {SL}(A)$ . The permutation group on r elements is denoted $\mathfrak S_r$ .

For at tensor $T\in A{\mathord { \otimes } } B{\mathord { \otimes } } C$ , define its symmetry group

(1) $$ \begin{align}G_T:=\{ (g_A,g_B,g_C)\in \operatorname{GL}(A)\times \operatorname{GL}(B)\times \operatorname{GL}(C)/(\mathbb C^*)^{\times 2}\mid (g_A,g_B,g_C)\cdot T=T\}. \end{align} $$

One quotients by $(\mathbb C^*)^{\times 2}:=\{(\lambda \operatorname {Id}_A,\mu \operatorname {Id}_B,\nu \operatorname {Id}_C)\mid \lambda \mu \nu =1\}$ because $(\nu \operatorname {Id}_A,\mu \operatorname {Id}_B,\frac 1{\nu \mu }\operatorname {Id}_C)$ is the kernel of the map $\operatorname {GL}(A)\times \operatorname {GL}(B)\times \operatorname {GL}(C){\mathord {\;\rightarrow \;}} \operatorname {GL}(A{\mathord { \otimes } } B{\mathord { \otimes } } C)$ . Lie algebras of Lie groups are denoted with corresponding symbols in old German script, for example, ${\mathfrak g}_T$ is the Lie algebra corresponding to  $G_T$ .

The Grassmannian of r planes through the origin is denoted $G(r,A)$ , which we will view in its Plücker embedding $G(r,A)\subset \mathbb P {\Lambda ^{r}} A$ . That is, given $E\in G(r,A)$ , that is, a linear subspace $E\subset A$ of dimension r, if is a basis of E, we represent E as a point in $\mathbb P({\Lambda ^{r}}A)$ by $[e_1\wedge \cdots \wedge e_r]$ . Here, the wedge product is defined by $e_1\wedge \cdots \wedge e_r:=\sum _{\sigma \in \mathfrak S_r}\mathrm {{sgn}}(\sigma ) e_{\sigma (1)}{\mathord { \otimes } } \cdots {\mathord { \otimes } } e_{\sigma (r)}$ .

For a set $Z\subset \mathbb P A$ , $\overline {Z}\subset \mathbb P A$ denotes its Zariski closure, $\hat Z\subset A$ denotes the cone over Z union the origin, $I(Z)=I(\hat Z)\subset \operatorname {Sym}(A^*)$ denotes the ideal of Z, that is, $I(Z)=\{ P\in \operatorname {Sym}(A^*)\mid P(z)=0 \forall z\in \hat Z\}$ , and $\mathbb C[\hat Z]=\operatorname {Sym}(A^*)/I(Z)$ , denotes the homogeneous coordinate ring of $\hat Z$ . Both $I(Z)$ and $\mathbb C[\hat Z]$ are $\mathbb {Z}$ -graded by degree.

We will be dealing with ideals on products of three projective spaces, that is, we will be dealing with polynomials that are homogeneous in three sets of variables, so our ideals with be $\mathbb {Z}^3$ -graded. More precisely, we will study ideals $I\subset \operatorname {Sym}(A^*){\mathord { \otimes } } \operatorname {Sym}(B^*){\mathord { \otimes } } \operatorname {Sym}(C^*)$ , and $I_{ijk}$ denotes the component in $S^iA^*{\mathord { \otimes } } S^jB^*{\mathord { \otimes } } S^kC^*$ .

Given $T\in A{\mathord { \otimes } } B{\mathord { \otimes } } C$ , we may consider it as a linear map $T_C: C^*{\mathord {\;\rightarrow \;}} A{\mathord { \otimes } } B$ , and we let $T(C^*)\subset A{\mathord { \otimes } } B$ denote its image and similarly for permuted statements. A tensor T is concise if the maps $T_A,T_B,T_C$ are injective, that is, if it requires all basis vectors in each of $A,B,C$ to write down in any basis.

We remark that the tensor T may be recovered up to isomorphism from any of the spaces $T(A^*),T(B^*),T(C^*)$ ; see, for example, [Reference Landsberg and Michałek33].

Elements $P\in S^iA^*{\mathord { \otimes } } S^jB^*{\mathord { \otimes } } S^k C^*$ may be viewed as differential operators on elements $X\in S^s A {\mathord { \otimes } } S^t B {\mathord { \otimes } } S^u C$ . Write for the contraction operation. The annihilator of X, denoted $\text {Ann}\,(X)$ , is defined to be the ideal of all $P\in \operatorname {Sym}(A^*){\mathord { \otimes } } \operatorname {Sym}(B^*){\mathord { \otimes } } \operatorname {Sym}(C^*)$ such that . In the case that $X=T\in A{\mathord { \otimes } } B{\mathord { \otimes } } C$ , its annihilator consists of all elements in degree $(i,j,k)$ with one of $i,j,k$ greater than one and the annihilators in low degrees are just the usual linear annihilators defined above. Explicitly, the annihilators in low degree are $T(C^*)^{\perp }\subset A^*{\mathord { \otimes } } B^*$ , $T(B^*)^{\perp }\subset A^*{\mathord { \otimes } } C^*$ and $T(A^*)^{\perp }\subset B^*{\mathord { \otimes } } C^*$ and $T^{\perp }\subset A^*{\mathord { \otimes } } B^*{\mathord { \otimes } } C^*$ .

2.2 Border rank decompositions as curves in Grassmannians

A border rank r decomposition of a tensor T is normally viewed as a curve $T(t)=\sum _{j=1}^r T_j(t)$ , where each $T_j(t)$ is rank one for all $t\neq 0$ , and $\lim _{t{\mathord {\;\rightarrow \;}} 0}T(t)=T$ . It will be useful to change perspective, viewing a border rank r decomposition of a tensor $T\in A{\mathord { \otimes } } B{\mathord { \otimes } } C$ as a curve $E_t\subset G(r,A{\mathord { \otimes } } B{\mathord { \otimes } } C)$ satisfying

1. (i) for all $t\neq 0$ , $E_t$ is spanned by r rank one tensors, and

2. (ii) $T\in E_0$ .

Example 2.1. The border rank decomposition

$$ \begin{align*} a_1{\mathord{ \otimes } } b_1{\mathord{ \otimes } } c_2+ a_1{\mathord{ \otimes } } b_2{\mathord{ \otimes } } c_1+a_2{\mathord{ \otimes } } b_1{\mathord{ \otimes } } c_1=\lim_{t{\mathord{\;\rightarrow\;}} 0}\frac 1t[(a_1+ta_2){\mathord{ \otimes } } (b_1+tb_2){\mathord{ \otimes } } (c_1+tc_2) - a_1{\mathord{ \otimes } } b_1{\mathord{ \otimes } } c_1] \end{align*} $$

may be rephrased as the curve

$$ \begin{align*} E_t &= [(a_1{\mathord{ \otimes } } b_1{\mathord{ \otimes } } c_1)\wedge (a_1+ta_2){\mathord{ \otimes } } (b_1+tb_2){\mathord{ \otimes } } (c_1+tc_2)] \\[2pt] &= \begin{aligned}[t] {[}(a_1{\mathord{ \otimes } } b_1{\mathord{ \otimes } } c_1)\wedge (&a_1{\mathord{ \otimes } } b_1{\mathord{ \otimes } } c_1+ t(a_1{\mathord{ \otimes } } b_1{\mathord{ \otimes } } c_2+ a_1{\mathord{ \otimes } } b_2{\mathord{ \otimes } } c_1+a_2{\mathord{ \otimes } } b_1{\mathord{ \otimes } } c_1) \\[2pt] {}& +t^2(a_1{\mathord{ \otimes } } b_2{\mathord{ \otimes } } c_2+ a_2{\mathord{ \otimes } } b_1{\mathord{ \otimes } } c_2+a_2{\mathord{ \otimes } } b_2{\mathord{ \otimes } } c_1) +t^3a_2{\mathord{ \otimes } } b_2{\mathord{ \otimes } } c_2) ] \end{aligned} \\[2pt] &= \begin{aligned}[t] {[}(a_1{\mathord{ \otimes } } b_1{\mathord{ \otimes } } c_1)\wedge ( & a_1{\mathord{ \otimes } } b_1{\mathord{ \otimes } } c_2+ a_1{\mathord{ \otimes } } b_2{\mathord{ \otimes } } c_1+a_2{\mathord{ \otimes } } b_1{\mathord{ \otimes } } c_1 \\[2pt] {}&+t (a_1{\mathord{ \otimes } } b_2{\mathord{ \otimes } } c_2+ a_2{\mathord{ \otimes } } b_1{\mathord{ \otimes } } c_2+a_2{\mathord{ \otimes } } b_2{\mathord{ \otimes } } c_1) +t^2a_2{\mathord{ \otimes } } b_2{\mathord{ \otimes } } c_2) ] \end{aligned} \\[2pt] &\subset G(2,A{\mathord{ \otimes } } B{\mathord{ \otimes } } C). \end{align*} $$

Here,

$$ \begin{align*}E_0=[(a_1{\mathord{ \otimes } } b_1{\mathord{ \otimes } } c_1)\wedge (a_1{\mathord{ \otimes } } b_1{\mathord{ \otimes } } c_2+ a_1{\mathord{ \otimes } } b_2{\mathord{ \otimes } } c_1+a_2{\mathord{ \otimes } } b_1{\mathord{ \otimes } } c_1)]. \end{align*} $$

2.3 Multigraded ideal associated to a border rank decomposition

Given a border rank r decomposition $T=\lim _{t{\mathord {\;\rightarrow \;}} 0}\sum _{j=1}^r T_j(t)$ , we have additional information. Let

$$ \begin{align*}I_t\subset \operatorname{Sym}(A^*){\mathord{ \otimes } } \operatorname{Sym}(B^*){\mathord{ \otimes } } \operatorname{Sym}(C^*) \end{align*} $$

denote the $\mathbb {Z}^3$ -graded ideal of the set of r distinct points $[T_1(t)]\cup \cdots \cup [T_r(t)]$ , where $I_{ijk,t}\subset S^iA^*{\mathord { \otimes } } S^jB^*{\mathord { \otimes } } S^kC^*$ . Sometimes, it is more convenient to work with $I_{ijk,t}^{\perp }$ which contains equivalent information.

Thus, in Example 2.1 above, $I_{ijk,t}^{\perp }=\langle a_1^i{\mathord { \otimes } } b_1^j{\mathord { \otimes } } c_1^k, (a_1+ta_2)^i{\mathord { \otimes } } (b_1+tb_2)^j{\mathord { \otimes } } (c_1+tc_2)^k\rangle $ , where the role of $a_2$ in Example 2.2 is played by $(a_1+ta_2)$ and similarly for $b_2,c_2$ . As $t{\mathord {\;\rightarrow \;}} 0$ , $I_{ijk,t}^{\perp }$ limits to $I_{ijk}^{\perp }=\langle a_1^i{\mathord { \otimes } } b_1^j{\mathord { \otimes } } c_1^k, ia_1^{i-1}a_2{\mathord { \otimes } } b_1^j{\mathord { \otimes } } c_1^k+ ja_1^i{\mathord { \otimes } } b_1^{j-1}b_2{\mathord { \otimes } } c_1^k+ k a_1^i{\mathord { \otimes } } b_1^j{\mathord { \otimes } } c_1^{k-1}c_2 \rangle $ .

If the r points are in general position, then $\text {codim}(I_{ijk,t})=r$ as long as $r\leq \operatorname {dim} S^iA^*{\mathord { \otimes } } S^jB^*{\mathord { \otimes } } S^kC^*$ ; see, for example, [Reference Buczyńska and Buczyński11, Lemma 3.9].

Let $\mathbf {a}\leq \mathbf {b}\leq \mathbf {c}$ . If $r\leq \binom {\mathbf {a}+1}2$ , then for all $(ijk)$ with $i+j+k>1$ , one has $r\leq \operatorname {dim} S^iA^*{\mathord { \otimes } } S^jB^*{\mathord { \otimes } } S^kC^*$ .

In all the examples in this paper $r\leq \binom {\mathbf {a}+1}2$ . For example, for $M_{\langle 2\mathbf {n}\mathbf {n}\rangle }$ , $\binom {\mathbf {a}+1}2=2\mathbf {n}^2+\mathbf {n}$ and we prove border rank lower bounds like $\mathbf {n}^2+1.32\mathbf {n}$ .

Thus, in this paper we may and will assume $\text {codim} (I_{ijk})=r$ for all $(ijk)$ with $i+j+k>1$ .

Thus, in addition to $E_0=I_{111,0}^{\perp }$ defined in §2.2, we obtain a limiting ideal I, where we define $I_{ijk}:=\lim _{t{\mathord {\;\rightarrow \;}} 0}I_{ijk,t}$ and the limit is taken in the Grassmannian of codimension r subspaces in $S^iA^*{\mathord { \otimes } } S^jB^*{\mathord { \otimes } } S^kC^*$ ,

$$ \begin{align*}G(\operatorname{dim}(S^iA^*{\mathord{ \otimes } } S^jB^*{\mathord{ \otimes } } S^kC^*)-r,S^iA^*{\mathord{ \otimes } } S^jB^*{\mathord{ \otimes } } S^kC^*). \end{align*} $$

We remark that there are subtleties here: The limiting ideal may not be saturated. See [Reference Buczyńska and Buczyński11] for a discussion.

Thus, we may assume a multigraded ideal I coming from a border rank r decomposition of a concise tensor T satisfies the following conditions:

1. (i) I is contained in the annihilator of T, which by definition says $I_{110}\subset T(C^*)^{\perp }$ , $I_{101}\subset T(B^*)^{\perp } $ , $I_{011}\subset T(A^*)^{\perp }$ and $I_{111}\subset T^{\perp }\subset A^*{\mathord { \otimes } } B^*{\mathord { \otimes } } C^*$ .

2. (ii) For all $(ijk)$ with $i+j+k>1$ , $\text {codim} I_{ijk}=r$ .

3. (iii) I is an ideal, so the multiplication maps

   (2) $$ \begin{align} I_{i-1,j,k}{\mathord{ \otimes } } A^*{\mathord{\,\oplus }\,} I_{i,j-1,k}{\mathord{ \otimes } } B^* {\mathord{\,\oplus }\,} I_{i,j,k-1}{\mathord{ \otimes } } C^*{\mathord{\;\rightarrow\;}} S^iA^*{\mathord{ \otimes } } S^jB^*{\mathord{ \otimes } } S^kC^* \end{align} $$

   have image contained in $I_{ijk}$ .

Here, equation (2) is the sum of three maps, the first of which is the restriction of the symmetrization map $S^{i-1}A^*{\mathord { \otimes } } A^*{\mathord { \otimes } } S^jB^*{\mathord { \otimes } } S^k C^*{\mathord {\;\rightarrow \;}} S^iA^*{\mathord { \otimes } } S^jB^*{\mathord { \otimes } } S^k C^*$ to $I_{i-1,j,k}{\mathord { \otimes } } A^*$ and similarly for the others. When $i-1=0$ , the first map is just the inclusion $I_{0jk}{\mathord { \otimes } } A^*\subset A^*{\mathord { \otimes } } S^jB^*{\mathord { \otimes } } S^kC^*$ .

One may prove border rank lower bounds for T by showing that for a given r, no such $I $ exists. For arbitrary tensors, we do not see any effective way to prove this, but for tensors with a large symmetry group, we have a vast simplification of the problem as described in the next subsection.

2.4 Lie’s theorem and consequences

Lie’s theorem may be stated as: Let H be a connected solvable group and W an H-module, then a closed H-fixed set $X\subset \mathbb P W$ contains an H-fixed point. Applying this fact to appropriately chosen sets X yield the Normal Form Lemma and its generalization, the Fixed Ideal Theorem.

By the Normal Form Lemma, in order to prove $\underline {\mathbf {R}}(T)>r$ , it is sufficient to rule out the existence of a border rank r decomposition $E_t$ where $E_0$ is a H-fixed point of $G(r, A{\mathord { \otimes } } B{\mathord { \otimes } } C)$ .

The conclusions of the theorems above are stronger for larger groups of symmetries H, so in this paper we will always fix a Borel subgroup ${\mathbb B}_T \subset G_T$ , that is, a maximal connected solvable subgroup of $G_T$ . Thus, we may assume a multigraded ideal I coming from a border rank r decomposition of T satisfies the additional condition:

1. (iv) Each $I_{ijk}$ is ${\mathbb B}_T$ -fixed.

As we explain in the next subsection, for the instances in considered in this paper, Borel fixed spaces are easy to list.

2.5 Borel fixed subspaces

All of the ${\mathbb B}_T$ -modules for which we would like to study ${\mathbb B}_T$ -fixed subspaces are also $G_T$ -modules, where $G_T$ is reductive. This fact simplifies the description of ${\mathbb B}_T$ -fixed subspaces, so we will assume this in what follows.

It will be convenient for us to linearize the problem by considering Lie algebras instead of Lie groups. Let ${\mathfrak g}_T$ be the Lie algebra of $G_T$ , and let $\mathfrak b_T \subset {\mathfrak g}_T$ be the Lie algebra of ${\mathbb B}_T\subset G_T$ . A subspace $S \subset M$ is ${\mathbb B}_T$ fixed if and only if it is $\mathfrak b_T$ fixed.

2.5.3 $\mathfrak {gl}_m$ and $\mathfrak {sl}_m$ weights

All of the groups appearing as $G_T$ in this paper are $\operatorname {GL}_m$ and $\operatorname {SL}_m$ and products of such. In this case, a Borel subgroup in some choice of basis is just the group of invertible upper triangular matrices (in the case of $\operatorname {SL}_m$ , with determinant one) or the product of such.

For ${\mathbb B}$ the invertible upper triangular matrices, $\mathfrak b$ is just all upper triangular matrices. Here, $\mathfrak b = {\mathfrak t} {\mathord {\,\oplus }\,} {\mathfrak n}$ , where ${\mathfrak t}$ is the diagonal matrices and ${\mathfrak n}$ is the set of upper triangular matrices with zero on the diagonal.

Let be the basis dual to the basis of ${\mathfrak t}\subset \mathfrak {gl}_m$ , and and write , where the $1$ is in the j-th slot. Let $\mathbb C^m$ have standard basis , with dual basis . Then are weight vectors of ${\mathfrak t}$ , and $e_j$ has weight $\epsilon _j$ .

If $g\in G$ acts on V by $v\mapsto gv$ , then the induced action on $V^*$ is $\alpha \mapsto \alpha \circ g{}^{-1}$ so that $g\cdot (\alpha (v))=( \alpha \circ g{}^{-1} )(g\cdot v)=\alpha (v)$ . When we differentiate this action, the induced Lie algebra action is $X.\alpha =-\alpha \circ X$ . Thus, considering the action of ${\mathfrak t}$ on $(\mathbb C^m)^*$ , are the set of weight vectors and .

For vectors in $(\mathbb C^m)^{{\mathord { \otimes } } d}$ , $wt(e_1^{{\mathord { \otimes } } a_1}{\mathord {\otimes \cdots \otimes } } e_m^{{\mathord { \otimes } } a_m})=a_1\epsilon _1+\cdots +a_m\epsilon _m$ and the weight is unchanged under permutations of the $d=a_1+\cdots +a_m$ factors. The partial order on weights of §2.5.1 may be written $a_1\epsilon _1 +\cdots + a_m\epsilon _m \geq b_1\epsilon _1 +\cdots + b_m\epsilon _m$ if $\sum _{i=1}^s a_{i}\geq \sum _{i=1}^s b_{i}$ for all s.

The Lie algebra $\mathfrak {sl}_m$ corresponding to $\operatorname {SL}_m$ consists of the trace free $m\times m$ matrices. Here, ${\mathfrak t}$ is the set of diagonal matrices with trace zero, so the set of weights is defined only modulo $\epsilon _1+\cdots +\epsilon _m$ . We will write $\mathfrak {sl}_m$ weights as $c_1\omega _1+\cdots +c_{m-1}\omega _{m-1}$ , where the $\omega _j:= \epsilon _1+\cdots +\epsilon _j$ are called the fundamental weights. Thus, in terms of $\mathfrak {sl}_m$ weights, $wt(e_1) = \omega _1$ , for $2\le s \le m-1$ , $wt(e_s) = \omega _s-\omega _{s-1}$ , $wt(e_m) = -\omega _{m-1}$ , and for all j, $wt(e^j) = -wt(e_j)$ .

In terms of $\mathfrak {sl}_m$ weights, the partial order is thus $a_1\omega _1+\cdots + a_{m-1}\omega _{m-1}\ge b_1\omega _1+\cdots + b_{m-1}\omega _{m-1}$ when $a_i\ge b_i$ for all i. For every $\mathfrak {sl}_m$ weight $\lambda \ge 0$ , there is a unique irreducible module denoted $V_{\lambda }$ containing a highest weight vector of weight $\lambda $ . See, for example, [Reference Humphreys26, Chap. 6] or [Reference Knapp28, Chap. 5, §2] for details.

Example 2.5 ( $\mathfrak {sl}_2$ as an $\mathfrak {sl}_2$ -module).

This example will be used in the proofs of Theorems 1.4 and 1.5. Figure 1 gives the weight diagram for $\mathfrak {sl}_2=\mathfrak {sl}(V)$ as a $\mathfrak {sl}_2$ -module under the adjoint action, that is, for $X,Y\in \mathfrak {sl}_2$ , $X.Y = XY-YX$ . Here, $v_1,v_2$ is a basis of V with dual basis $v^1,v^2$ and $v^i_j:=v_j{\mathord { \otimes } } v^i$ .

Figure 1 Weight diagram for adjoint representation of $\mathfrak {sl}_2$

The only ${\mathbb B}$ -fixed subspaces are $0$ , $ \langle v_1^2\rangle $ , $ \langle v_1^2, v_1^1- v_2^2\rangle $ and $ \langle v_1^2, v_1^1- v_2^2, v^2_1\rangle $ .

Example 2.6 ( $\mathfrak {sl}_3$ as an $\mathfrak {sl}_3$ -module).

This example will be used in the proofs of Theorems 1.1 and 1.6. Figure 2 gives the weight diagram for $\mathfrak {sl}_3$ as an $\mathfrak {sl}_3$ -module under the adjoint action. As above ${v^i_j=v_j{\mathord { \otimes } } v^i}$ . The oval is around the two-dimensional weight zero subspace, which has four distinguished one-dimensional subspaces: the images of the two raising operators in and the kernels of the two raising operators out. Additional arrows indicating these relationships have been added to the weight diagram.

Figure 2 Weight diagram for adjoint representation of $\mathfrak {sl}_3$

The ${\mathbb B}$ -fixed subspaces of dimension three are $ \langle v_1^3, v_2^3, v_1^2\rangle $ , $ \langle v_1^3, v_2^3,2v_3^3-(v_1^1+v_2^2)\rangle $ and $ \langle v_1^3, v_1^2,2v_1^1-(v_2^2+v_3^3)\rangle $ .

The ${\mathbb B}$ -fixed subspaces of dimension four are a family parametrized by $[s,t]\in \mathbb P^1$ : $ \langle v_1^3, v_2^3, v^2_1, s(v^2_2-v^3_3)+t(v^1_1-v^2_2) \rangle $ . There are no others: We cannot include the entire weight zero space, as then we must also include all the positive weight vectors for a total dimension of five, exceeding our limit. If we include a negative weight vector, we must include its image in the weight zero space, which again raises to all positive weight vectors, exceeding our limit.

The ${\mathbb B}$ -fixed subspaces of dimension five are $\langle v_1^3, v_2^3, v^2_1,v^2_2-v^3_3,v^2_3\rangle $ , $\langle v_1^3, v_2^3, v^2_1,v^1_1-v^2_2,v^1_2\rangle $ and the span of the weight $\geq 0$ space $\langle v_1^3, v_2^3, v^2_1,v^2_2-v^3_3,v^1_1-v^2_2\rangle $ . This is easy to see as were $v^1_2,v^2_3$ both present we would need the full weight zero space making the dimension six, and $v^1_3$ can be included only if the whole Lie algebra is included.

Example 2.7 (Bilinear forms on $U^*$ ).

This example will be used in the proof of Theorem 1.2. Let $\operatorname {dim} U=3$ with basis $u_1,u_2,u_3$ . Figure 3 gives the weight diagram for $U{\mathord { \otimes } } U=S^2U{\mathord {\,\oplus }\,} {\Lambda ^{2}} U$ as an $\mathfrak {sl}(U)$ -module. The action of $X\in \mathfrak {sl}(U)$ on a matrix $Z\in U{\mathord { \otimes } } U$ is $Z\mapsto XZ+ZX^{\mathbf {t}}$ . There are two ${\mathbb B}$ -fixed lines $\langle (u_1)^2\rangle $ and $\langle u_1\wedge u_2\rangle $ , there is a $1$ -(projective) parameter $[s,t]\in {\mathbb P^{1}}$ space of ${\mathbb B}$ -fixed $2$ -planes, $\langle (u_1)^2, su_1u_2+tu_1\wedge u_2\rangle $ plus an isolated one $\langle u_1\wedge u_2,u_1\wedge u_3\rangle $ .

Figure 3 Weight diagram for $U{\mathord { \otimes } } U$ when $U= \mathbb C^3$ . There are six distinct weights appearing, indicated on the right. On the far left are the weight vectors in $S^2U$ , and in the middle are the weight vectors in ${\Lambda ^{2}}U$

Example 2.8 (Tensor products of modules for different groups).

Suppose M and N are modules for groups G and H, respectively. Then $M{\mathord {\otimes }}N$ is a $G\times H$ module with weight spaces $M_{\lambda }{\mathord { \otimes } } N_{\mu }$ , as $\lambda $ and $\mu $ range over all pairs of weights of M and N. For each arrow $M_{\lambda }\to M_{\nu }$ in the weight diagram of M corresponding to the raising operator x, there is an edge $M_{\lambda } {\mathord { \otimes } } N_{\mu } \to M_{\nu }{\mathord { \otimes } } N_{\mu }$ corresponding to the raising operator $x {\mathord {\,\oplus }\,} 0 \in {\mathfrak g} {\mathord {\,\oplus }\,} {\mathfrak h}$ . Similarly, for each arrow $N_{\mu }\to N_{\nu }$ in the weight diagram of N corresponding to the raising operator y, there is an edge $M_{\lambda } {\mathord { \otimes } } N_{\mu } \to M_{\lambda }{\mathord { \otimes } } N_{\nu }$ corresponding to the raising operator $0 {\mathord {\,\oplus }\,} y \in {\mathfrak g} {\mathord {\,\oplus }\,} {\mathfrak h}$ . Example 2.9 is a special case of this applied twice to obtain the diagram of a triple tensor product $U^*{\mathord { \otimes } } \mathfrak {sl}(V){\mathord { \otimes } } W$ with $\mathbf {u}=\mathbf {v}=\mathbf {w}=2$ . This example with $\mathbf {u}=\mathbf {w}=\mathbf {n}$ and $\mathbf {v}=2$ (resp. $\mathbf {v}=3$ ) is used in the proof of Theorem 1.5 (resp. 1.6).

Example 2.9 ( $U^*{\mathord { \otimes } } \mathfrak {sl}(V){\mathord { \otimes } } W$ as an $\mathfrak {sl}(U)\times \mathfrak {sl}(V)\times \mathfrak {sl}(W)$ -module).

This example is crucial for the case of $M_{\langle \mathbf {u},\mathbf {v},\mathbf {w}\rangle }$ as then $A{\mathord { \otimes } } B=(U^*{\mathord { \otimes } } V){\mathord { \otimes } } (V^*{\mathord { \otimes } } W)= U^*{\mathord { \otimes } } \mathfrak {sl}(V){\mathord { \otimes } } W{\mathord {\,\oplus }\,} M_{\langle \mathbf {u},\mathbf {v},\mathbf {w}\rangle }(C^*)$ . When $U,V,W$ each have dimension two, Figure 4 gives the $\mathfrak {sl}(U)\times \mathfrak {sl}(V)\times \mathfrak {sl}(W)$ -weight diagram for $U^*{\mathord { \otimes } } \mathfrak {sl}(V){\mathord { \otimes } } W$ . Set ${x^i_j=u^i{\mathord { \otimes } } v_j}$ and $y^i_j=v^i{\mathord { \otimes } } w_j$ . There is a unique ${\mathbb B}$ -fixed line, $\langle x^2_1{\mathord { \otimes } } y^2_1\rangle $ , three ${\mathbb B}$ -fixed $2$ -planes, $\langle x^2_1{\mathord { \otimes } } y^2_1, x^1_1{\mathord { \otimes } } y^2_1\rangle $ , $\langle x^2_1{\mathord { \otimes } } y^2_1, x^2_1{\mathord { \otimes } } y^2_2\rangle $ , and $\langle x^2_1{\mathord { \otimes } } y^2_1, x^2_1{\mathord { \otimes } } y^1_1-x^2_2{\mathord { \otimes } } y^2_1\rangle $ , and four ${\mathbb B}$ -fixed $3$ -planes, $\langle x^2_1{\mathord { \otimes } } y^2_1, x^1_1{\mathord { \otimes } } y^2_1, x^2_1{\mathord { \otimes } } y^1_1-x^2_2{\mathord { \otimes } } y^2_1\rangle $ , $\langle x^2_1{\mathord { \otimes } } y^2_1,x^2_1{\mathord { \otimes } } y^1_1-x^2_2{\mathord { \otimes } } y^2_1, x^2_1{\mathord { \otimes } } y^2_2\rangle $ , $\langle x^2_1{\mathord { \otimes } } y^2_1, x^1_1{\mathord { \otimes } } y^2_1, x^2_1{\mathord { \otimes } } y^2_2\rangle $ , and $\langle x^2_1{\mathord { \otimes } } y^2_1, x^2_1{\mathord { \otimes } } y^1_1-x^2_2{\mathord { \otimes } } y^2_1, x^2_2{\mathord { \otimes } } y^1_1\rangle $ .

Figure 4 Weight diagram for $U^*{\mathord { \otimes } } \mathfrak {sl}(V){\mathord { \otimes } } W$ when $U=V=W=\mathbb C^2$ . Left are the weight vectors and right the weights: Since $\mathfrak {sl}_2$ weights are just $j\omega _1$ , we have just written $(i\,|\,j\,|\,k)$ for the $\mathfrak {sl}(U){\mathord {\,\oplus }\,} \mathfrak {sl}(V){\mathord {\,\oplus }\,} \mathfrak {sl}(W)$ weight. Raisings in $U^*$ correspond to NW (northwest) arrows, those in W to NE (northeast) arrows and those in $\mathfrak {sl}(V)$ to upward arrows

6.1 Refinement of the $(210)$ test for matrix multiplication

Recall $A=U^*{\mathord { \otimes } } V$ , $B=V^*{\mathord { \otimes } } W$ , $C=W^*{\mathord { \otimes } } U$ and $M_{\langle \mathbf {u},\mathbf {v},\mathbf {w}\rangle }=\operatorname {Id}_U{\mathord { \otimes } } \operatorname {Id}_V{\mathord { \otimes } } \operatorname {Id}_W$ and the notation $\omega _j$ for the fundamental $\mathfrak {sl}$ -weights from §2.5. Let $V_{\mu }$ denote the irreducible $\mathfrak {sl}(V)$ -module with highest weight $\mu $ . We have the following decompositions as $\operatorname {SL}(U)\times \operatorname {SL}(V)$ -modules: (note $V_{\omega _2+\omega _{\mathbf {v}-1}}$ does not appear when $\mathbf {v}=2$ , and when $\mathbf {v}=3$ , $V_{\omega _2+\omega _{\mathbf {v}-1}}=V_{2\omega _2}$ ):

(9) $$ \begin{align} {\Lambda^{2}} (U^*{\mathord{ \otimes } } V){\mathord{ \otimes } } V^*&= \begin{aligned}[t] &(S^2U^*{\mathord{ \otimes } } V_{\omega_1}){\mathord{\,\oplus }\,} ({\Lambda^{2}} U^*{\mathord{ \otimes } } V_{\omega_1}) \\ &{\mathord{\,\oplus }\,} (S^2U^*{\mathord{ \otimes } } V_{\omega_2+\omega_{\mathbf{v}-1}}){\mathord{\,\oplus }\,} ({\Lambda^{2}} U^*{\mathord{ \otimes } } V_{2\omega_1+\omega_{\mathbf{v}-1}}), \end{aligned} \end{align} $$

(10) $$ \begin{align} S^2(U^*{\mathord{ \otimes } } V){\mathord{ \otimes } } V^*&= \begin{aligned}[t] &(S^2U^*{\mathord{ \otimes } } V_{2\omega_1+\omega_{\mathbf{v}-1}}){\mathord{\,\oplus }\,} ({\Lambda^{2}} U^*{\mathord{ \otimes } } V_{\omega_2+\omega_{\mathbf{v}-1}}) \\ {}&{\mathord{\,\oplus }\,}(S^2U^*{\mathord{ \otimes } } V_{\omega_1}) {\mathord{\,\oplus }\,} ({\Lambda^{2}} U^*{\mathord{ \otimes } } V_{\omega_1} ), \end{aligned} \end{align} $$

(11) $$ \begin{align} A{\mathord{ \otimes } } M_{\langle \mathbf{u},\mathbf{v},\mathbf{w}\rangle}(C^*)&= \begin{aligned}[t] &(U^*{\mathord{ \otimes } } V){\mathord{ \otimes } } (U^*{\mathord{ \otimes } } \operatorname{Id}_V{\mathord{ \otimes } } W) \\ &= (S^2U^*{\mathord{ \otimes } } V_{\omega_1}{\mathord{ \otimes } } W){\mathord{\,\oplus }\,} ({\Lambda^{2}} U^*{\mathord{ \otimes } } V_{\omega_1} {\mathord{ \otimes } } W), \end{aligned} \end{align} $$

(12) $$ \begin{align} V{\mathord{ \otimes } } \mathfrak {sl}(V)&=V_{ \omega_1 }{\mathord{\,\oplus }\,} V_{2\omega_1+\omega_{\mathbf{v}-1}}{\mathord{\,\oplus }\,} V_{\omega_2+\omega_{\mathbf{v}-1}}, \end{align} $$

(13) $$ \begin{align} (U^*{\mathord{ \otimes } } V){\mathord{ \otimes } } (U^*{\mathord{ \otimes } } \mathfrak {sl}(V)) &= \begin{aligned}[t] &(S^2U^*{\mathord{ \otimes } } V_{2\omega_1+\omega_{\mathbf{v}-1}}){\mathord{\,\oplus }\,} ({\Lambda^{2}} U^*{\mathord{ \otimes } } V_{2\omega_1+\omega_{\mathbf{v}-1}}) \\ {}&{\mathord{\,\oplus }\,}(S^2U^*{\mathord{ \otimes } } V_{\omega_1} ) {\mathord{\,\oplus }\,} ({\Lambda^{2}} U^*{\mathord{ \otimes } } V_{\omega_1} ) \\ {}&{\mathord{\,\oplus }\,} (S^2U^*{\mathord{ \otimes } } V_{\omega_2+\omega_{\mathbf{v}-1}}) {\mathord{\,\oplus }\,} ({\Lambda^{2}} U^*{\mathord{ \otimes } } V_{\omega_2+\omega_{\mathbf{v}-1}}). \end{aligned} \end{align} $$

These formulas follow from the following basic formulas: for any vector spaces $U,V$ , one has the $\operatorname {GL}(U)\times \operatorname {GL}(V)$ decompositions $S^2(U{\mathord { \otimes } } V)=S^2U{\mathord { \otimes } } S^2V{\mathord {\,\oplus }\,} {\Lambda ^{2}} U{\mathord { \otimes } } {\Lambda ^{2}} V$ ; see, for example, [Reference Landsberg30, §2.7.1] and the decomposition ${\Lambda ^{2}} (U{\mathord { \otimes } } V)= S^2U{\mathord { \otimes } } {\Lambda ^{2}} V{\mathord {\,\oplus }\,} S^2 U{\mathord { \otimes } } {\Lambda ^{2}} V$ is derived similarly, and one has the $\operatorname {GL}(U)$ decomposition $U{\mathord { \otimes } } U=S^2U{\mathord {\,\oplus }\,} {\Lambda ^{2}} U$ . Finally, the Pieri formula (see, e.g., [Reference Fulton and Harris20, §6.1, eqns 6.8,6.9]) gives $S^2V{\mathord { \otimes } } V^*=V_{2\omega _1}{\mathord { \otimes } } V_{\omega _{\mathbf {v}-1}}=V_{2\omega _1+ \omega _{\mathbf {v}-1}} {\mathord {\,\oplus }\,} V_{\omega _1}$ and ${\Lambda ^{2}} V{\mathord { \otimes } } V^*=V_{\omega _2}{\mathord { \otimes } } V_{\omega _{\mathbf {v}-1}}=V_{\omega _2+\omega _{\mathbf {v}-1}}{\mathord {\,\oplus }\,} V_{\omega _1}$ .

Note that $V_{\omega _1}$ is isomorphic to V and

$$\begin{align*}\textstyle \operatorname{dim}(V_{2\omega_1+\omega_{\mathbf{v}-1}}) = \frac {1}{2} \mathbf{v}^3+ \frac{1} {2} \mathbf{v}^2 -\mathbf{v} , \ \ \ \operatorname{dim}( V_{\omega_2+\omega_{\mathbf{v}-1}} )=\frac {1}{2}\mathbf{v}^3-\frac{1}{ 2}\mathbf{v}^2-\mathbf{v}. \end{align*}$$

Proposition 6.1. Write $E_{110}:=M_{\langle \mathbf {u},\mathbf {v},\mathbf {w}\rangle }(C^*){\mathord {\,\oplus }\,} E_{110}'$ , where $E_{110}'\subset U^*{\mathord { \otimes } } \mathfrak {sl}(V){\mathord { \otimes } } W$ . The dimension of the kernel of the map (7) $E_{110}{\mathord { \otimes } } A {\mathord {\;\rightarrow \;}} {\Lambda ^{2}} A{\mathord { \otimes } } B$ equals the dimension of the kernel of the skew symmetrization followed by projection map

(14) $$ \begin{align} E_{110}'{\mathord{ \otimes } } A {\mathord{\;\rightarrow\;}} S^2U^*{\mathord{ \otimes } } V_{\omega_2+\omega_{\mathbf{v}-1}}{\mathord{ \otimes } } W{\mathord{\,\oplus }\,} {\Lambda^{2}} U^*{\mathord{ \otimes } } V_{2\omega_1+\omega_{\mathbf{v}-1}}{\mathord{ \otimes } } W, \end{align} $$

and the kernel of equation (14) is

(15) $$ \begin{align} (E_{110}'{\mathord{ \otimes } } A )\cap [ U^{*{\mathord{ \otimes } } 2}{\mathord{ \otimes } } V_{\omega_1}{\mathord{ \otimes } } W{\mathord{\,\oplus }\,} S^2U^*{\mathord{ \otimes } } V_{2\omega_1+\omega_{\mathbf{v}-1}}{\mathord{ \otimes } } W{\mathord{\,\oplus }\,} {\Lambda^{2}} U^*{\mathord{ \otimes } } V_{\omega_2+\omega_{\mathbf{v}-1}}{\mathord{ \otimes } } W]. \end{align} $$

Proof. Write M for the target of equation (14). We have the following commutative diagram, where horizontal arrows form exact sequences:

The bottom row reflects the decomposition (9) tensored with W. The middle vertical arrow is the skew symmetrization map (7), and since it is the restriction of a $\operatorname {GL}(U)\times \operatorname {GL}(V)\times \operatorname {GL}(W)$ equivariant map, by Schur’s lemma, its submodule $(U^*{\mathord { \otimes } } \operatorname {Id}_V{\mathord { \otimes } } W) {\mathord { \otimes } } A = (U^*{\mathord { \otimes } } V) {\mathord { \otimes } } (U^*{\mathord { \otimes } } \operatorname {Id}_V{\mathord { \otimes } } W)$ must have image contained in $(U^*)^{{\mathord { \otimes } } 2}{\mathord { \otimes } } V_{\omega _1}{\mathord { \otimes } } W$ . The induced right vertical arrow is the map (14).

We show the left vertical arrow is an isomorphism, from which the claim on the kernel dimension of equation (14) will follow by, for example, the snake lemma. We have the decomposition into irreducible modules

$$\begin{align*}(U^*{\mathord{ \otimes } } V){\mathord{ \otimes } } (U^*{\mathord{ \otimes } } \operatorname{Id}_V{\mathord{ \otimes } } W)=S^2U^*{\mathord{ \otimes } } V{\mathord{ \otimes } } \operatorname{Id}_V{\mathord{ \otimes } } W{\mathord{\,\oplus }\,} {\Lambda^{2}} U^*{\mathord{ \otimes } } V{\mathord{ \otimes } } \operatorname{Id}_V{\mathord{ \otimes } } W. \end{align*}$$

The vertical left arrow is an equivariant map, so by Schur’s lemma, it is sufficient to see that a single vector in each of the modules on the right has nonzero image. We check the highest weight vectors:

$$ \begin{align*} &(u^{\mathbf{u}} {\mathord{ \otimes } } v_1){\mathord{ \otimes } } u^{\mathbf{u}}{\mathord{ \otimes } } (\sum_j v_j{\mathord{ \otimes } } v^j){\mathord{ \otimes } } w_1 \mapsto \sum_{\rho>1} (u^{\mathbf{u}}{\mathord{ \otimes } } v_1)\wedge (u^{\mathbf{u}}{\mathord{ \otimes } } v_{\rho}) {\mathord{ \otimes } } v^{\rho}{\mathord{ \otimes } } w_1, \ \mathrm{and} \\ & \begin{aligned}[t] \big[(u^{\mathbf{u}} {\mathord{ \otimes } } v_1){\mathord{ \otimes } } u^{\mathbf{u}-1} - {}&(u^{\mathbf{u}-1} {\mathord{ \otimes } } v_1){\mathord{ \otimes } } u^{\mathbf{u}}\big ] {\mathord{ \otimes } } (\sum_j v_j{\mathord{ \otimes } } v^j){\mathord{ \otimes } } w_1 \mapsto \\ &\sum_j \big[(u^{\mathbf{u}}{\mathord{ \otimes } } v_1)\wedge (u^{\mathbf{u}-1}{\mathord{ \otimes } } v_j) -(u^{\mathbf{u}-1}{\mathord{ \otimes } } v_1)\wedge (u^{\mathbf{u}}{\mathord{ \otimes } } v_j)\big]{\mathord{ \otimes } } v^j{\mathord{ \otimes } } w_1 \end{aligned} \end{align*} $$

Now, equation (14) is a restriction of the surjective equivariant map $ U^*{\mathord { \otimes } } \mathfrak {sl}(V) {\mathord { \otimes } } W {\mathord { \otimes } } A\to M$ . Comparing modules in the irreducible decompositions of the source and target in view of equation (13), we obtain that equation (15) is the kernel of equation (14).

6.2 $ \underline {\mathbf {R}}(M_{\langle 2\rangle })>6$ revisited

In this case, the map (14) takes image in ${\Lambda ^{2}} U^*{\mathord { \otimes } } S^2V{\mathord { \otimes } } V^*{\mathord { \otimes } } W$ . We have the following images.

For the highest weight vector $x^2_1{\mathord { \otimes } } y^2_1$ times the four basis vectors of A (with their $\mathfrak {sl}(V)$ -weights in the second column), the image of equation (14) is spanned by

$$ \begin{align*} x^1_1\wedge x^2_1{\mathord{ \otimes } } y^2_1 & \quad 3\omega_1\\ x^1_2\wedge x^2_1{\mathord{ \otimes } } y^2_1 & \quad \omega_1. \end{align*} $$

(Note, e.g., $x^2_2{\mathord { \otimes } } x^2_1{\mathord { \otimes } } y^2_1$ maps to zero under the skew-symmetrization map as $u^2{\mathord { \otimes } } u^2$ projects to zero in ${\Lambda ^{2}} U^*$ .) For $A{\mathord { \otimes } } (x^2_1{\mathord { \otimes } } y^1_1-x^2_2{\mathord { \otimes } } y^2_1)$ (the lowering of $x^2_1{\mathord { \otimes } } y^2_1$ under $\mathfrak {sl}(V)$ ), the image is spanned by

$$ \begin{align*} x^1_1\wedge (x^2_1{\mathord{ \otimes } } y^1_1-x^2_2{\mathord{ \otimes } } y^2_1) & \quad \omega_1\\ x^1_2\wedge (x^2_1{\mathord{ \otimes } } y^1_1-x^2_2{\mathord{ \otimes } } y^2_1) & \quad {-\omega_1}. \end{align*} $$

Since W has nothing to do with the map, we don’t need to compute the image of, for example, $A{\mathord { \otimes } } x^2_1{\mathord { \otimes } } y^2_2$ to know its contribution to the kernel, as it must be the same dimension as that of $A{\mathord { \otimes } } x^2_1{\mathord { \otimes } } y^2_1$ , just with a different W-weight.

Were $\underline {\mathbf {R}}(M_{\langle 2\rangle })=6$ , $E_{110}'$ would have dimension two, spanned by the highest weight vector and one lowering of it, and in order to be a candidate, its image in ${\Lambda ^{2}} U^*{\mathord { \otimes } } S^3V{\mathord { \otimes } } W$ would have to have dimension at most two. Taking $E_{110}'=\langle x^2_1{\mathord { \otimes } } y^2_1, x^2_1{\mathord { \otimes } } y^1_1-x^2_2{\mathord { \otimes } } y^2_1\rangle $ , the image of equation (14) has dimension three. Taking $E_{110}'=\langle x^2_1{\mathord { \otimes } } y^2_1, x^2_1{\mathord { \otimes } } y^2_2\rangle $ , the image of equation (14) has dimension four. Finally, taking $E_{110}'=\langle x^1_1{\mathord { \otimes } } y^2_1, x^2_1{\mathord { \otimes } } y^2_1\rangle $ , by transpose symmetry (see §4), the image of the $(120)$ -version of equation (14) must have dimension four and we conclude.

7.1 Overview

To prove border rank lower bounds for a fixed tensor using border apolarity, one checks a list of candidates for components of a multigraded ideal. It is not immediate how to extend the technique to sequences of tensors in $\mathbf {n}$ . Even in good situations such as in Theorems 1.5 and 1.6 where there are large Borel subgroups, candidate components can still occur in positive-dimensional families, and there is an exponential growth in $\mathbf {n}$ in the number of families to check. We overcome this problem by introducing several new ideas.

We restrict attention to only the (110)-graded ideal component and the application of the dual form of the (120) and (210) tests of §3. For each given candidate component, we forget everything about it except for the dimensions of certain internal weight spaces. We then analyze the kernels of Proposition 3.5 as sums of “local” contributions from each internal weight space. As we consider only dimension information, we determine upper bounds on the contributions. At this point, there are still many discrete cases of possible choices of these internal dimensions to consider. We use techniques from convex optimization to show that the relevant kernel contributions for any choice is no better than a constant more than that of a small fixed number of choices. We call this step the “globalization”. These choices can then be completely analyzed as functions of $\mathbf {n}$ .

7.2 Preliminaries

Recall that in these proofs $\mathbf {u}=\mathbf {w}=\mathbf {n}$ and $\mathbf {v}$ is $2$ or $3$ . Let $E_{110}' \subset U^* {\mathord { \otimes } } \mathfrak {sl}(V) {\mathord { \otimes } } W$ be a ${\mathbb B}$ -fixed subspace. In particular, $E_{110}'$ is fixed under the torus of $\operatorname {GL}(U)\times \operatorname {GL}(W)$ , so we may write $E_{110}' = \bigoplus _{s,t} u^{\mathbf {n}-s+1}{\mathord { \otimes } } X_{s,t} {\mathord { \otimes } } w_t$ , where $X_{s,t}\subset \mathfrak {sl}(V)$ . Since $E_{110}'$ is fixed under the Borel subgroups of $\operatorname {GL}(U)$ , $\operatorname {GL}(V)$ and $\operatorname {GL}(W)$ , for each s and t we have

1. 1. $X_{s,t}\subset \mathfrak {sl}(V)$ is ${\mathbb B}_V\subset \operatorname {GL}(V)$ fixed,

2. 2. $X_{s,t} \supset X_{s+1,t}$ and

3. 3. $X_{s,t}\supset X_{s,t+1}$ .

Define the outer structure of $E_{110}'$ to be the data

$$ \begin{align*}(s,t,\operatorname{dim} X_{s,t}), \ \ 0\leq s,t\leq\mathbf{n}. \end{align*} $$

Define the inner structure at site $(s,t)$ to be $X_{s,t}$ .

We may consider the outer structure of $E_{110}'$ as an $\mathbf {n}\times \mathbf {n}$ grid, with each grid point $(s,t)$ labelled by the dimension of the corresponding $X_{s,t}$ . We will represent such filled grids by the corresponding Young diagrams on the nonzero labels so that the upper left box corresponds with the highest weight. Here, labels weakly decrease going to the right and down. It is reasonable to imagine such a filled Young diagram rotated $45^{\circ }$ clockwise to put the highest weight at the top, as in the corresponding weight diagram (see Example 2.9, where $\mathbf {n}=\mathbf {v}=2$ ).

In the case of $\mathfrak {sl}_2$ , each $X_{s,t}$ is determined by its dimension, so an outer structure completely specifies a corresponding $E_{110}'$ . In the case of $\mathfrak {sl}_3$ , information about the particular inner structures is lost passing from $E_{110}'$ to its outer structure.

Example 7.1. Here are three examples with $\mathbf {v}=2$ and $\rho =4$ .

The diagram corresponds to

$$\begin{align*}E_{110}' = \langle u^{\mathbf{n}}{\mathord{ \otimes } } v^2{\mathord{ \otimes } } v_1{\mathord{ \otimes } } w_1, u^{\mathbf{n}}{\mathord{ \otimes } } v^2{\mathord{ \otimes } } v_1{\mathord{ \otimes } } w_2, u^{\mathbf{n}}{\mathord{ \otimes } } v^2{\mathord{ \otimes } } v_1{\mathord{ \otimes } } w_3, u^{\mathbf{n}-1}{\mathord{ \otimes } } v^2{\mathord{ \otimes } } v_1{\mathord{ \otimes } } w_1\rangle. \end{align*}$$

The diagram corresponds to

$$\begin{align*}E_{110}' = \langle u^{\mathbf{n}}{\mathord{ \otimes } } v^2{\mathord{ \otimes } } v_1{\mathord{ \otimes } } w_1, u^{\mathbf{n}}{\mathord{ \otimes } } (v^1{\mathord{ \otimes } } v_1-v^2{\mathord{ \otimes } } v_2){\mathord{ \otimes } } w_1, u^{\mathbf{n}}{\mathord{ \otimes } } v^2{\mathord{ \otimes } } v_1{\mathord{ \otimes } } w_2, u^{\mathbf{n}-1}{\mathord{ \otimes } } v^2{\mathord{ \otimes } } v_1{\mathord{ \otimes } } w_1\rangle. \end{align*}$$

The diagram corresponds to

$$\begin{align*}E_{110}' = \langle u^{\mathbf{n}}{\mathord{ \otimes } } v^2{\mathord{ \otimes } } v_1{\mathord{ \otimes } } w_1, u^{\mathbf{n}}{\mathord{ \otimes } } (v^1{\mathord{ \otimes } } v_1-v^2{\mathord{ \otimes } } v_2){\mathord{ \otimes } } w_1, u^{\mathbf{n}}{\mathord{ \otimes } } v^1{\mathord{ \otimes } } v_2{\mathord{ \otimes } } w_1, u^{\mathbf{n}}{\mathord{ \otimes } } v^2{\mathord{ \otimes } } v_1{\mathord{ \otimes } } w_2\rangle. \end{align*}$$

The transpose symmetry discussed in §4 maps $ E_{110}' = \bigoplus _{s,t} u^{\mathbf {n}-s+1}{\mathord { \otimes } } X_{s,t} {\mathord { \otimes } } w_t$ to $\bigoplus _{s,t} u^{\mathbf {n}-t+1}{\mathord { \otimes } } X_{s,t}^{\mathbf {t}} {\mathord { \otimes } } w_s$ , that is, the inner structure at site $(s,t)$ becomes the transpose of the inner structure at site $(t,s)$ . In particular, transpose symmetry conjugates the diagram corresponding to the outer structure. In view of this symmetry, it is sufficient to study the $(210)$ test only, for then everything we can say is also a statement about the $(120)$ test under this transpose.

As mentioned above, we split the calculation of the kernel into a local and global computation. We bound the local contribution to the kernel at site $(s,t)$ by a function of $s,t$ and $\operatorname {dim} X_{s,t}$ . Once this is done, the theorems are proved by solving the resulting combinatorial optimization problem over outer structures.

Recall the expression (15) and let K denote the term in brackets, that is,

(16) $$ \begin{align} \begin{aligned} K &= (U^*)^{{\mathord{ \otimes } } 2}{\mathord{ \otimes } } V_{\omega_1}{\mathord{ \otimes } } W{\mathord{\,\oplus }\,} S^2U^* {\mathord{ \otimes } } V_{2\omega_1+\omega_{\mathbf{v}-1}}{\mathord{ \otimes } } W{\mathord{\,\oplus }\,} {\Lambda^{2}} U^*{\mathord{ \otimes } } V_{\omega_2+\omega_{\mathbf{v}-1}}{\mathord{ \otimes } } W \\ &\subset (U^*)^{{\mathord{ \otimes } } 2} {\mathord{ \otimes } } V{\mathord{ \otimes } } \mathfrak {sl}(V) {\mathord{ \otimes } } W. \end{aligned} \end{align} $$

We may filter $E_{110}'$ by ${\mathbb B}$ -fixed subspaces such that each quotient corresponds to the inner structure contribution over some site $(s,t)$ . Call such a filtration admissible. Let $\Sigma _1\subset \Sigma _2\subset \cdots \subset \Sigma _f= E_{110}'$ be an admissible filtration, and put

(17) $$ \begin{align} K_g = (\Sigma_g {\mathord{ \otimes } } A )\cap K. \end{align} $$

Then the dimension of equation (15) may be written as the sum over g of $ \mathrm {dim}\; (K_g/K_{g-1})$ , and we may upper bound the dimension of equation (15) by upper bounding each $\mathrm {dim}\; (K_g/K_{g-1})$ . We obtain bounds on $\mathrm {dim}\; (K_g/K_{g-1})$ which depend only on s and $j=j_g := \mathrm {dim}\; (\Sigma _g / \Sigma _{g-1})$ . For $\mathfrak {sl}_2$ , this is Lemma 7.2, and for $\mathfrak {sl}_3$ , this is Lemma 7.4. As discussed above, bounds on the kernel of the $(120)$ map are obtained by symmetry; specifically, the bound is the same with s replaced by t.

Once these lemmas are established, the claims on fixed finite values of $\mathbf {n}$ may be immediately settled by enumerating the finitely many possible outer structures and checking that none gives a large enough kernel for both the $(210)$ and $(120)$ maps. The claims on infinite sequences of $\mathbf {n}$ require us to work more carefully, and we prove the required bounds on the solution to such problems parameterized by $\mathbf {n}$ in Lemma 7.7.

7.3 The local argument

Lemma 7.2. Let $\operatorname {dim} V=2$ , $\operatorname {dim} U=\mathbf {n}$ . Fix an admissible filtration such that $ \Sigma _g\subset E_{110}'$ contains the $\mathfrak {sl}(V)$ -subspace at site $(s,t)$ and $\Sigma _{g-1}$ does not. Write j for the dimension of the $\mathfrak {sl}(V)$ -subspace at site $(s,t)$ . Then

$$ \begin{align*}\operatorname{dim} (K_g/ K_{g-1})=a_js+b_j, \end{align*} $$

where $a_j,b_j$ are the following functions of j:

Lemma 7.2 is proved later this section.

Lemma 7.4. Let $\operatorname {dim} V=3$ , $\operatorname {dim} U=\mathbf {n}$ . Fix an admissible filtration such that $ \Sigma _g\subset E_{110}'$ contains the $\mathfrak {sl}(V)$ -subspace at site $(s,t)$ and $\Sigma _{g-1}$ does not. Write j for the dimension of the $\mathfrak {sl}(V)$ -subspace at site $(s,t)$ . Then

$$ \begin{align*}\operatorname{dim} (K_g/ K_{g-1})\leq a_js+b_j, \end{align*} $$

where $a_j,b_j$ are the following functions of j:

In order to prove Lemmas 7.2 and 7.4, we first observe the following.

Proposition 7.5. The included module $V_{\omega _1} \subset V{\mathord { \otimes } } \mathfrak {sl}(V)$ has weight basis

$$ \begin{align*}\overline{v}_i := \sum_{j\neq i} [\mathbf{v} v_j{\mathord{ \otimes } } (v_i{\mathord{ \otimes } } v^j)- v_i{\mathord{ \otimes } } (v_j{\mathord{ \otimes } } v^j)] + (\mathbf{v}-1) v_i{\mathord{ \otimes } } v_i{\mathord{ \otimes } } v^i, \ \ 1\leq i\leq \mathbf{v}. \end{align*} $$

Proof of Lemmas 7.2 and 7.4.

We begin in somewhat greater generality, not fixing $\mathbf {v}=\operatorname {dim} V$ . We must bound $\mathrm {dim}\; K_g - \mathrm {dim}\; K_{g-1}$ , where $K_g$ is given by equation (17). Write $X\subset \mathfrak {sl}(V)$ for the inner structure at $(s,t)$ so that $\Sigma _g=\Sigma _{g-1} \oplus u^{\mathbf {n}-s+1} {\mathord { \otimes } } X {\mathord { \otimes } } w_t$ . Write

$$ \begin{align*}V_0 &= \varnothing,\\ V_1 &= V_{\omega_1}, \\ V_2& = V_{\omega_1} {\mathord{\,\oplus }\,} V_{2\omega_1+\omega_{\mathbf{v}-1}}, \\ V_3 &= V_{\omega_1} {\mathord{\,\oplus }\,} V_{2\omega_1+\omega_{\mathbf{v}-1}} {\mathord{\,\oplus }\,} V_{\omega_2+\omega_{\mathbf{v}-1}} = V{\mathord{ \otimes } } \mathfrak {sl}(V). \end{align*} $$

Note that $V_2 = V_3$ when $\mathbf {v} = 2$ . Then $\{V_f\}_f$ is a (partial) flag for $V{\mathord { \otimes } } \mathfrak {sl}(V)$ , and

$$ \begin{align*} S_f := U^*{\mathord{ \otimes } } U^{*(s-1)}{\mathord{ \otimes } } V_3 {\mathord{ \otimes } } W + U^{*{\mathord{ \otimes } } 2} {\mathord{ \otimes } } V_f {\mathord{ \otimes } } W + U^{*{\mathord{ \otimes } } 2}{\mathord{ \otimes } } V_3 {\mathord{ \otimes } } W_{(t-1)} \end{align*} $$

is a flag for $U^{*{\mathord { \otimes } } 2}{\mathord { \otimes } } V_3{\mathord { \otimes } } W$ , where we have written $U^{*(s)} = \mathrm {span}\{u^{\mathbf {n}},\ldots ,u^{\mathbf {n}-s+1}\}$ and $W_{(t )} = \mathrm {span}\{w_1,\ldots , w_t\}$ . Hence, $S_f\cap K_g$ is a flag for $K_g$ with $S_3\cap K_g = K_g$ and $S_0\cap K_g \subseteq K_{g-1}$ . The fact that the inclusion $S_0\cap K_g\subseteq K_{g-1}$ may be strict is the only place in this argument we prove an inequality rather than equality. Use the isomorphism of quotient vector spaces

(18) $$ \begin{align} \frac{K_g\cap S_f}{K_g\cap S_{f-1}} = \frac{(K_g\cap S_f) + S_{f-1}}{S_{f-1}} \end{align} $$

to obtain the successive quotients of $\{S_f \cap K_g\}_f$ as subspaces of

(19) $$ \begin{align} \frac{U^{*{\mathord{ \otimes } } 2}{\mathord{ \otimes } } V_3{\mathord{ \otimes } } W }{ S_{f-1}}=\frac{U^{*{\mathord{ \otimes } } 2}} { U^*{\mathord{ \otimes } } U^{*(s-1)}} \otimes \frac{V_3}{V_{f-1}} \otimes \frac{W}{W_{(t-1)} }. \end{align} $$

Write $K^f$ for the f-th summand of equation (16) so that $K \cap S_f = K^f + K\cap S_{f-1}$ . Intersecting with $\Sigma _g{\mathord { \otimes } } A$ and adding $S_{f-1}$ , we obtain

$$ \begin{align*}K_g\cap S_f + S_{f-1} &= (K^f + S_{f-1}) \cap (\Sigma_g {\mathord{ \otimes } } A) + S_{f-1}\\ & = (K^f + S_{f-1}) \cap (U^*{\mathord{ \otimes } } u^{\mathbf{n}-s+1}{\mathord{ \otimes } } V{\mathord{ \otimes } } X {\mathord{ \otimes } } w_t + S_{f-1}). \end{align*} $$

We may now pass in each side of the intersection to the right-hand side of equation (19), after which the intersection may be computed term by term. To compute the intersection in the $U^{*{\mathord { \otimes } } 2} /(U^*{\mathord { \otimes } } U^{*(s-1)})$ term, momentarily write $\overline {Z} = Z + U^*{\mathord { \otimes } } U^{*(s-1)}$ for $Z\in U^{*{\mathord { \otimes } } 2}$ and observe that

$$ \begin{align*}\overline{S^2 U^*} \cap \overline{ U^*{\mathord{ \otimes } } u^{\mathbf{n}-s+1}} = \overline{U^{*s} {\mathord{ \otimes } } u^{\mathbf{n}-s+1}}, \ \mathrm{and} \ \overline{{\Lambda^{2}} U^*} \cap \overline{U^*{\mathord{ \otimes } } u^{\mathbf{n}-s+1}} = \overline{U^{*(s-1)} {\mathord{ \otimes } } u^{\mathbf{n}-s+1}}. \end{align*} $$

Therefore, the right-hand side of equation (18) may be written, for $f=1$ , $2$ and $3$ respectively,

$$ \begin{gather*} U^*\otimes (u^{\mathbf{n}-s+1} + U^{*(s-1)}) \otimes [(V\otimes X)\cap V_1 ]\otimes (w_t + W_{(t-1)})\\ U^{*s}\otimes (u^{\mathbf{n}-s+1} + U^{*(s-1)}) \otimes [(V\otimes X + V_1) \cap V_2 ]\otimes (w_t + W_{(t-1)})\\ U^{*(s-1)}\otimes (u^{\mathbf{n}-s+1} + U^{*(s-1)}) \otimes [V\otimes X + V_2 ]\otimes (w_t + W_{(t-1)}). \end{gather*} $$

Write

$$ \begin{align*}Y &= (V{\mathord{ \otimes } } X)\cap V_1,\\ Y' &= ((V{\mathord{ \otimes } } X + V_1)\cap V_2)/ V_1, \mathrm{\ and }\\ Y" &= (V{\mathord{ \otimes } } X + V_2)/ V_2, \end{align*} $$

and write their dimensions, respectively, as $\mathbf {y},\mathbf {y}',\mathbf {y}"$ . We obtain

$$ \begin{align*}\mathrm{dim}\; K_g =\mathrm{dim}\; (S_0\cap K_g) + \mathbf{y}\mathbf{n} + \mathbf{y}'s + \mathbf{y}"(s-1)\leq \mathrm{dim}\; K_{g-1} + \mathbf{y}\mathbf{n} + \mathbf{y}'s + \mathbf{y}"(s-1), \end{align*} $$

the sum of the successive quotient dimensions of $\{S_f\cap K_g\}_f$ .

Thus, when $j=\mathbf {v}^2 - 1$ , that is, $X=\mathfrak {sl}(V)$ , the desired result follows from $\mathbf {y} = \mathbf {v}$ , $\mathbf {y}' = \mathrm {dim}\; V_{2\omega _{1}+\omega _{\mathbf {v}-1}}$ and $\mathbf {y}" = \mathrm {dim}\; V_{\omega _{2}+\omega _{\mathbf {v}-1}}$ .

In all cases, Y has a basis consisting of weight vectors and is closed under raising operators. Hence, by Proposition 7.5, $Y = \mathrm {span} \{\overline {v}_i\mid i\le \mathbf {y} \}$ .

Consider the case $j=\mathbf {v}^2-2$ , that is X is the span of all weight vectors of $\mathfrak {sl}(V)$ except $v_{\mathbf {v}} {\mathord { \otimes } } v^1$ . Then $\overline {v}_{\mathbf {v}}$ is not an element of Y because in the monomial basis, the monomial $v_1{\mathord { \otimes } } (v_{\mathbf {v}}{\mathord { \otimes } } v^1)$ fails to have a nonzero coefficient in any element of Y. Hence, $\mathbf {y} \le \mathbf {v} -1$ , and the trivial $\mathbf {y}' \le \mathrm {dim}\; V_{2\omega _{1}+\omega _{\mathbf {v}-1}}$ , and $\mathbf {y}" \le \mathrm {dim}\; V_{\omega _{2}+\omega _{\mathbf {v}-1}}$ give the asserted upper bounds.

By similar reasoning, when $\mathbf {v} =3$ , considering Example 2.6, we obtain the bounds $\mathbf {y} = 0$ when $j=1,2$ and $\mathbf {y} \le 1$ when $j=3,4,5,6$ . For all values of j except $1$ , the result then follows from

(20) $$ \begin{align} \mathrm{dim}\; K_g - \mathrm{dim}\; K_{g-1} &\leq (j\mathbf{v} - \mathbf{y})s + \mathbf{y} \mathbf{n} - \mathbf{y}"\\ &\nonumber \le (j\mathbf{v} - \mathbf{y})s + \mathbf{y}\mathbf{n}, \end{align} $$

as $\mathbf {y}+\mathbf {y}'+\mathbf {y}" = j\mathbf {v}$ . The upper bound for $\mathbf {v} =2$ , $j=1$ , is settled similarly.

We must argue more for the $j=1$ upper bound for $\mathbf {v} =3$ , namely that $\mathbf {y}" \ge 2$ . For this, consider

$$ \begin{align*}V{\mathord{ \otimes } } \mathfrak {sl}(V) {\mathord{\,\oplus }\,} V_{\omega_1} = V{\mathord{ \otimes } } V {\mathord{ \otimes } } V^* = S^2 V {\mathord{ \otimes } } V^* {\mathord{\,\oplus }\,} {\Lambda^{2}} V {\mathord{ \otimes } } V^*\ \mathrm{and}\ {\Lambda^{2}} V {\mathord{ \otimes } } V^* = V_{\omega_2+\omega_{\mathbf{v}-1}} {\mathord{\,\oplus }\,} V_{\omega_1}. \end{align*} $$

Because we have $\mathbf {y} = 0$ , the dimension $\mathbf {y}"$ of the projection of $V{\mathord { \otimes } } X$ onto $V_{\omega _2+\omega _{\mathbf {v}-1}}$ is the same as that onto ${\Lambda ^{2}} V {\mathord { \otimes } } V^*$ . We have the images $v_2\wedge v_1{\mathord { \otimes } } v^{3}$ and $v_3\wedge v_1{\mathord { \otimes } } v^{3}$ of $v_2{\mathord { \otimes } } v_1{\mathord { \otimes } } v^3$ and $v_3{\mathord { \otimes } } v_1{\mathord { \otimes } } v^3$ , respectively, whence $\mathbf {y}" \ge 2$ as required.

To see the upper bounds in the $\mathbf {v} = 2$ cases are sharp, note that in this case $V_{\omega _2 + \omega _{\mathbf {v}-1}} = \varnothing $ , so $\mathbf {y}"=0$ . The $j=1$ case is thus automatic from equation (20), and for $j=2$ , we must show $\mathbf {y} \ge 1$ . In this case, however, we have $\overline {v}_1 = 2v_2 {\mathord { \otimes } }(v_1{\mathord { \otimes } } v^2) + v_1{\mathord { \otimes } } (v_1{\mathord { \otimes } } v^1 - v_2{\mathord { \otimes } } v^2) \in V{\mathord { \otimes } } X$ , as required.

In Lemma 7.7 below, the linear functions of s in Lemmas 7.2 and 7.4 appear as $a_{\mu _{s,t}}s+b_{\mu _{s,t}}$ .

7.4 The globalization

Write $\mu $ for a Young diagram filled with nonnegative integer labels. The label in position $(s,t)$ is denoted $\mu _{s,t}$ , and sums over $s,t$ are to be taken over the boxes of $\mu $ . As before, each $\mu $ will correspond to a possible outer structure.

We remark that the lemmas in this section and the next may be used for $M_{\langle \mathbf {m}\mathbf {n}\mathbf {n}\rangle }$ for any $\mathbf {n}\geq \mathbf {m}$ .

The following lemma allows us to reduce from considering all possible outer structures and the corresponding bounds on the dimension of the kernels of the $(210)$ and $(120)$ tests to considering three (resp. eight) possible kernel dimensions in the case of $\mathbf {v}=2$ (resp. $\mathbf {v}=3$ ).

Lemma 7.7. Fix $k\in \mathbb N$ , $0 \le a_1 \le \cdots \le a_k$ , and $b_i \in \mathbb R$ , $1\le i \le k$ . Let $\mu $ be a Young diagram filled with labels in the set $\{1,\ldots , k\}$ , nonincreasing in rows and columns. Write $\rho = \sum _{s,t} \mu _{s,t}$ . Then

(21) $$ \begin{align} \min \Big\{ \sum_{s,t} a_{\mu_{s,t}}s + b_{\mu_{s,t}} , \sum_{s,t} a_{\mu_{s,t}}t + b_{\mu_{s,t}} \Big\} \le \max_{1\le j\le k} \bigg\{ \frac{a_j \rho^2}{8j^2} + (a_j+b_j) \frac{\rho}{j} \bigg\}. \end{align} $$

Lemma 7.7 is proved in §7.5.

Proof of Theorem 1.5(3).

Let $E_{110}'\subset U^*{\mathord { \otimes } } \mathfrak {sl}(V){\mathord { \otimes } } W$ be a ${\mathbb B}$ -fixed subspace, and let $\mu $ be the corresponding outer structure. We apply Lemma 7.7 with $k=3$ and $a_i$ and $b_i$ from Lemma 7.2 to obtain an upper bound on the smaller of the kernel dimensions of the $(120)$ and $(210)$ maps. The resulting upper bound is $\max \{\frac {1}{4} \rho ^2 + 2\rho , \frac {3}{32} \rho ^2 + \frac {3+\mathbf {n}}{2} \rho , \frac {1}{18} \rho ^2 + \frac {4+2\mathbf {n}}{3} \rho \}$ .

Fix $\epsilon> 0$ . We must show that if $\rho = (3\sqrt {6} - 6 - \epsilon ) \mathbf {n}$ , then each of $\frac {1}{4} \rho ^2 + 2\rho $ , $\frac {3}{32} \rho ^2 + \frac {3+\mathbf {n}}{2} \rho $ , and $ \frac {1}{18} \rho ^2 + \frac {4+2\mathbf {n}}{3} \rho $ is strictly smaller than $\mathbf {n}^2 + \rho $ . Substituting and solving for $\mathbf {n}$ , we obtain that this holds for the last expression when

$$\begin{align*}\mathbf{n}> \frac{6}{\epsilon} \frac{3\sqrt{6}+6 - \epsilon}{6\sqrt{6} - \epsilon}, \end{align*}$$

and when $\epsilon < \frac {1}{4} $ , this condition implies the other two inequalities.

Proof of Theorem 1.6.

Proceeding in the same way as in the proof of Theorem 1.5(3), we apply Lemma 7.7 with $\mu $ the outer structure corresponding to an arbitrary ${\mathbb B}$ -fixed subspace $E_{110}' \subset U^*{\mathord { \otimes } } \mathfrak {sl}(V){\mathord { \otimes } } W$ , $k=8$ , and $a_i$ and $b_i$ corresponding to the inner structure contribution upper bounds obtained in Lemma 7.4. We obtain the smaller of the kernel dimensions of the $(120)$ and $(210)$ maps is at most the largest of the following,

Now, if one takes $\rho =\lfloor \sqrt {\frac 83}\mathbf {n}\rfloor $ , the kernel upper bound for each j is strictly less than $\mathbf {n}^2+\rho $ . This follows for $j=1$ because $\sqrt {\frac 83}\mathbf {n}$ is irrational. This follows for $2\le j\le 8$ because $\mathbf {n}\geq 18$ . Hence, at least one of the kernels of the (120) and (210) maps is too small, and $\underline {\mathbf {R}}(M_{\langle 3\mathbf {n}\mathbf {n}\rangle })> \mathbf {n}^2 + \rho $ , as required.

7.5 Proof of Lemma 7.7

We will reduce Lemma 7.7 to the following, which may be viewed as a continuous reformulation. Its proof depends on a delicate perturbation argument.

Lemma 7.9. Fix $k\in \mathbb N$ , $c_i \ge 0$ , $d_i \in \mathbb R$ , for $1\le i\le k$ . Write $C_j = \sum _{i=1}^j c_i$ and $D_j = \sum _{i=1}^j d_i$ . For all choices of $x_i$ , $y_j$ satisfying the constraints $x_1 \ge \cdots \ge x_k \ge 0$ , $y_1 \ge \cdots \ge y_k \ge 0$ , and $\sum _i x_i + y_i = \rho $ , the following inequality holds:

(22) $$ \begin{align} \min \Big\{ \sum_{i\le k} c_ix_i^2 + d_i(x_i+y_i), \sum_{i\le k} c_iy_i^2 + d_i(x_i+y_i) \Big\} \le \max_{1\le j\le k} \bigg\{ \frac{\rho^2}{4j^2} C_j + \frac{\rho}{j} D_j \bigg\}. \end{align} $$

Proof. As both the left- and right-hand sides are continuous in the $c_i$ , it suffices to prove the result under the assumption $c_i> 0$ . The idea of the proof is the following: Any choice of $x_i$ and $y_i$ which has at least two degrees of freedom inside its defining polytope can be perturbed in such a way that the local linear approximations to the two polynomials on the left -hand side do not decrease; that is, two closed half planes in $\mathbb R^2$ containing $(0,0)$ also intersect aside from $(0,0)$ . Each polynomial on the left strictly exceeds its linear approximation at any point, and thus one can strictly improve the left-hand side with a perturbation. The case of at most one degree of freedom is settled directly.

Write $x_{k+1}= y_{k+1} =0$ , and define $x_i' = x_i- x_{i+1}$ and $y_i' = y_i-y_{i+1}$ so that $x_i = \sum _{j=i}^k x_j'$ and $y_i = \sum _{j=i}^k y_j'$ . Then $x_i', y_i' \ge 0$ and $\sum _{i=1}^k i (x_i' + y_i') = \rho $ . Suppose at least three of the $ x_i' $ , $ y_j' $ are nonzero, we will show the expression on the left-hand side of equation (22) is not maximal. Write three of the nonzero $ x_i',y_j'$ as $\overline x,\overline y,\overline z$ . Replace them by $\overline x+\epsilon _1$ , $\overline y+\epsilon _2$ , $\overline z+\epsilon _3$ , with the $\epsilon _i$ to be determined. This will preserve the equation $\sum _i x_i+y_i=\rho $ only if $\epsilon _1+\epsilon _2+\epsilon _3=0$ , so we require this. Substitute these values into

$$ \begin{align*}E_L:=\sum_{i\le k} c_ix_i^2 + d_i(x_i+y_i)\ \mathrm{and}\ E_R:=\sum_{i\le k} c_iy_i^2 + d_i(x_i+y_i). \end{align*} $$

View $E_L,E_R$ as polynomial expressions in the $\epsilon _j$ . Then

$$ \begin{align*}E_L=\sum_i c_i S_{L,i}^2 + L_L + d, \ E_R=\sum_i c_i S_{R,i}^2 + L_R + d, \end{align*} $$

where $S_{L,i},S_{R,i}$ and $L_L,L_R$ are linear forms in the $\epsilon _i$ , and $d\in \mathbb R$ . Each $S_{L,i},S_{R,i}$ is a sum of some subset of the $\epsilon _i$ , and the union of them span the $2$ -plane $\langle \epsilon _1,\epsilon _2,\epsilon _3\rangle /\langle \sum \epsilon _j=0 \rangle $ . Consider the linear map $T = L_L \oplus L_R : \langle \epsilon _1,\epsilon _2,\epsilon _3\rangle /\langle \sum \epsilon _j=0 \rangle \to \mathbb R^2$ . If T is nonsingular, then for any $\epsilon>0$ , there are constants $\overline \epsilon _j$ , with $\sum \overline \epsilon _j=0$ so that $T(\overline \epsilon _1,\overline \epsilon _2,\overline \epsilon _3) = (\epsilon ,\epsilon )$ , and it is possible to choose $\epsilon $ so that $ \overline x + \overline \epsilon _1, \overline y + \overline \epsilon _2, \overline z +\overline \epsilon _3 \ge 0$ . Then this new assignment strictly improves the old one. Otherwise, if T is singular, then there is an admissible $(\overline \epsilon _1,\overline \epsilon _2,\overline \epsilon _3) \ne 0$ in the kernel of T, where again we may assume the same nonnegativity condition. The corresponding assignment does not change $L_L,L_R$ , but as the $S_{L,i},S_{R,i}$ span the linear forms, at least one them is nonzero. Consequently, at least one of the modified $E_L,E_R$ is strictly larger after the perturbation, and neither is smaller. If, say, only $E_L$ is strictly larger, and $ x_i'> 0$ , we may substitute $ x_i' - \epsilon $ and $ y_i' + \epsilon $ for $ x_i' $ and $ y_i' $ for some $\epsilon>0$ to make both $E_L$ and $E_R$ strictly larger.

Thus, the left-hand side is maximized at an assignment where at most two of $x_i'$ and $y_i'$ are nonzero. It is clear that at least one of each of $x_i'$ and $y_i'$ must be nonzero, so there is exactly one of each, say $x_s' = \alpha $ and $y_t' = \beta $ . It is clear at the maximum that $ \sum _{i\le k} c_ix_i^2 + d_i(x_i+y_i) = \sum _{i\le k} c_iy_i^2 + d_i(x_i+y_i) $ , from which it follows that $ \alpha ^2 C_s = \sum _{i\le k} c_ix_i^2 =\sum _{i\le k} c_iy_i^2 = \beta ^2 C_t $ and $\alpha \sqrt {C_s} = \beta \sqrt {C_t}$ . We also have $s\alpha + t\beta = \rho $ . Notice that

$$\begin{align*}\alpha = \frac{\rho \sqrt{C_t}}{s\sqrt{C_t} + t\sqrt{C_s}},\quad \beta = \frac{\rho \sqrt{C_s}}{s\sqrt{C_t} + t\sqrt{C_s}} \end{align*}$$

satisfy the equations so that the optimal value obtained is

$$\begin{align*}\sum_{i\le k} c_ix_i^2 + d_i(x_i+y_i) = \alpha^2 C_s + \alpha D_s + \beta D_t = \frac{\rho}{s\sqrt{C_t} + t\sqrt{C_s}} \left( \frac{\rho C_s C_t}{s\sqrt{C_t} + t\sqrt{C_s}} + \sqrt{C_t}D_s + \sqrt{C_s} D_t \right). \end{align*}$$

By the arithmetic mean-harmonic mean inequality, we have

$$\begin{align*}\frac{ \rho C_s C_t}{s\sqrt{C_t} + t\sqrt{C_s}} = \frac{ \rho}{ \frac{s}{C_s \sqrt{C_t}} + \frac{t}{C_t \sqrt{C_s}}} \le \frac{ \rho}{4} \bigg[ \frac{C_s \sqrt{C_t}}{s} + \frac{C_t \sqrt{C_s}}{t} \bigg] \end{align*}$$

so that

$$ \begin{align*} \frac{\rho C_s C_t}{s\sqrt{C_t} + t\sqrt{C_s}} + \sqrt{C_t}D_s + \sqrt{C_s} D_t &\le \frac{ \rho}{4} \bigg[ \frac{C_s \sqrt{C_t}}{s} + \frac{C_t \sqrt{C_s}}{t} \bigg] + \sqrt{C_t}D_s + \sqrt{C_s} D_t \\ &= \frac{s\sqrt{C_t}+t\sqrt{C_s}}{\rho} \left[\frac{s\alpha} {\rho} \Big( \frac{\rho^2}{4s^2} C_s + \frac{\rho}{s} D_s \Big) + \frac{t\beta} {\rho}\Big( \frac{\rho^2}{4t^2} C_t + \frac{\rho}{t} D_t \Big) \right] \\ &\le \frac{s\sqrt{C_t}+t\sqrt{C_s}}{\rho} \max \bigg\{ \frac{\rho^2}{4s^2} C_s + \frac{\rho}{s} D_s, \frac{\rho^2}{4t^2} C_t + \frac{\rho}{t} D_t \bigg\}, \end{align*} $$

with the last inequality holding because $ \frac {s\alpha }{\rho } + \frac {t\beta }{\rho } = 1$ . Multiplying both sides by $ \frac {\rho }{s\sqrt {C_t} + t\sqrt {C_s}} $ , we conclude.

We prove one final lemma on partitions that will enable the reduction of Lemma 7.7 to an instance of Lemma 7.9.

For a partition , write $ \ell (\lambda )=q$ and define

$$\begin{align*}n(\lambda) := \sum_i (i-1) \lambda_i. \end{align*}$$

Let $\lambda '$ denote the conjugate partition.

Proof. We prove the result by induction on $\lambda _1=\ell (\lambda ')$ . When $\ell (\lambda ') = 1$ , we have

$$\begin{align*}\textstyle n(\lambda) = \binom{\lambda^{\prime}_1}{2} = \frac{1}{2} (\lambda^{\prime}_1 - \frac{1}{2} )^2 - \frac{1}{8} = \frac{1}{8} ( \lvert{\lambda}\rvert + \lambda^{\prime}_1 - \lambda_1)^2 - \frac{1}{8}, \end{align*}$$

as required. Now, assume $k = \ell (\lambda ')> 1$ . Write $\mu $ for the partition where $\ell (\mu ') = k - 1$ and $\mu ^{\prime }_i = \lambda ^{\prime }_i$ , $i\le k-1$ . If $\lambda = (3,3)$ , we are done by direct calculation; hence, otherwise we may assume the result holds for $\mu $ by the induction hypothesis.

$$ \begin{align*} \textstyle n(\lambda) & \textstyle = n(\mu) + \binom{\lambda^{\prime}_k}{2} \\ & \textstyle \le \frac{1}{8}(\lvert\mu\rvert +\mu_1' - \mu_1 )^2 - \frac{1}{8} + \binom{\lambda^{\prime}_k}{2} \\ & \textstyle = \frac{1}{8}(\lvert \lambda\rvert - \lambda^{\prime}_k +\lambda_1' - (\lambda_1 - 1) )^2 - \frac{1}{8} + \frac{1}{2} \lambda_k' (\lambda_k'-1) \\ & \textstyle = \frac{1}{8}(\lvert\lambda\rvert +\lambda_1' - \lambda_1)^2 - \frac{1}{8} - \frac{1}{4} ( \lvert\lambda\rvert +\lambda_1' - \lambda_1 - \frac{5}{2} \lambda^{\prime}_k + \frac{1}{2}) (\lambda^{\prime}_k - 1) \end{align*} $$

We must show $\frac {1}{4} ( \lvert \lambda \rvert +\lambda _1' - \lambda _1 - \frac {5}{2} \lambda ^{\prime }_k + \frac {1}{2}) (\lambda ^{\prime }_k - 1)\geq 0$ . If $\lambda _k' = 1$ , this is immediate; otherwise, we show the first factor is nonnegative. We have $\lvert \lambda \rvert -\lambda _1 \ge k\lambda _k' - k$ , so

$$\begin{align*}\textstyle \lvert\lambda\rvert +\lambda_1' - \lambda_1 - \frac{5}{2} \lambda^{\prime}_k + \frac{1}{2} \ge (\lambda_1' - \lambda_k') + \frac{2k-3}{2} (\lambda_k'-1) - 1. \end{align*}$$

If $k =2$ , then by assumption $\lambda _1' \ge 3$ , and considering separately the cases $\lambda _2' = 2$ and $\lambda _2' \ge 3$ yields the result. Otherwise $k\ge 3$ , and because $\lambda _k' \ge 2$ , we again conclude.

Proof of Lemma 7.7.

For each $1\le i\le k$ , let $\lambda ^i$ be the partition corresponding to the boxes of $\mu $ labeled $\ge i$ . Write $a_0 = b_0 = 0$ . Then

(23) $$ \begin{align} \textstyle \sum_{s,t} a_{\mu_{s,t}}s + b_{\mu_{s,t}} \nonumber &= \textstyle \sum_{s,t} \sum_{i=1}^{\mu_{s,t}} (a_i-a_{i-1})s + b_i-b_{i-1} \\ \nonumber &= \textstyle \sum_{i=1}^k \sum_{s,t \in \lambda^i} (a_i-a_{i-1})s + b_i-b_{i-1} \\ \nonumber &= \textstyle \sum_{i=1}^k (a_i - a_{i-1}) n(\lambda^i) + (a_i - a_{i-1} + b_i - b_{i-1})\lvert{\lambda^i}\rvert \\ &\le \textstyle \sum_{i=1}^k \left[\frac{1}{2} (a_i-a_{i-1})\right] \left(\frac{1}{2} (\lvert{\lambda^i}\rvert + (\lambda^i)^{\prime}_1- \lambda^i_1)\right)^2 + \left[a_i - a_{i-1} + b_i - b_{i-1}\right]\lvert{\lambda^i}\rvert \end{align} $$

where we have used Lemma 7.11 to obtain the last inequality. Set

$$ \begin{align*} c_i &= \textstyle\frac{1}{2} (a_i-a_{i-1})\\ d_i &=\textstyle a_i - a_{i-1} + b_i - b_{i-1}\\ x_i &=\textstyle \frac{1}{2} (\lvert{\lambda^i}\rvert + (\lambda^i)^{\prime}_1- \lambda^i_1)\\ y_i &=\textstyle \frac{1}{2} (\lvert{\lambda^i}\rvert - (\lambda^i)^{\prime}_1+ \lambda^i_1). \end{align*} $$

Then equation (23) becomes

$$ \begin{align*}\sum_{i=1}^k c_i x_i^2 + d_i (x_i + y_i). \end{align*} $$

Similarly, $ \sum _{s,t} a_{\mu _{s,t}}t + b_{\mu _{s,t}} \le \sum _{i=1}^k c_i y_i^2 + d_i (x_i + y_i) $ . Now, $\sum _i x_i + y_i = \sum _i \lvert {\lambda ^i}\rvert = \rho $ and the $x_i$ and $y_i$ are each nonnegative and nonincreasing. Hence, by Lemma 7.9,

$$\begin{align*}\min \Big\{ \sum_{s,t} a_{\mu_{s,t}}s + b_{\mu_{s,t}}, \sum_{s,t} a_{\mu_{s,t}}t + b_{\mu_{s,t}} \Big\} = \max_{1\le j\le k} \bigg\{ \frac{a_j \rho^2}{8j^2} + (a_j+b_j) \frac{\rho}{j} \bigg\}, \end{align*}$$

as required.