# Universal points in the asymptotic spectrum of tensors

The asymptotic restriction problem for tensors is to decide, given tensors s and t, whether the nth tensor power of s can be obtained from the (n+o(n))th tensor power of t by applying linear maps to the tensor legs (this we call restriction), when n goes to infinity. In this context, Volker Strassen, striving to understand the complexity of matrix multiplication, introduced in 1986 the asymptotic spectrum of tensors. Essentially, the asymptotic restriction problem for a family of tensors X, closed under direct sum and tensor product, reduces to finding all maps from X to the reals that are monotone under restriction, normalised on diagonal tensors, additive under direct sum and multiplicative under tensor product, which Strassen named spectral points. Strassen created the support functionals, which are spectral points for oblique tensors, a strict subfamily of all tensors. Universal spectral points are spectral points for the family of all tensors. The construction of nontrivial universal spectral points has been an open problem for more than thirty years. We construct for the first time a family of nontrivial universal spectral points over the complex numbers, using quantum entropy and covariants: the quantum functionals. In the process we connect the asymptotic spectrum to the quantum marginal problem and to the entanglement polytope. To demonstrate the asymptotic spectrum, we reprove (in hindsight) recent results on the cap set problem by reducing this problem to computing asymptotic spectrum of the reduced polynomial multiplication tensor, a prime example of Strassen. A better understanding of our universal spectral points construction may lead to further progress on related questions. We additionally show that the quantum functionals are an upper bound on the recently introduced (multi-)slice rank.

## Authors

• 9 publications
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## 1 Introduction

### 1.1 The asymptotic restriction problem

We study the asymptotic restriction problem, following the pioneering work of Volker Strassen [Str86, Str87, Str88, Str91]. The asymptotic restriction problem is a problem about multilinear maps over an arbitrary field . Letting be the standard basis of , one may equivalently think of  as the -tensor defined by where goes over . To state the asymptotic restriction problem we need the concepts restriction and tensor product. Let and be multilinear maps. We say restricts to , and write , if there are linear maps such that where denotes composition. We naturally define the tensor product as the multilinear map defined by . We say  restricts asymptotically to , written , if there is a sequence of natural numbers such that

 f⊗n+a(n)≥g⊗nwhenn→∞.

The asymptotic restriction problem is: given and , decide whether .

Applications of the asymptotic restriction problem include computing the computational complexity of matrix multiplication in algebraic complexity theory [AS81, Blä01, BI11, Lan14, Str69, CW90, Sto10, Wil12, LG14, CU03, CKSU05] (see also [BCS97, Lan12, Blä13, Lan17]), deciding the feasibility of an asymptotic transformation between pure quantum states via stochastic local operations and classical communication (slocc) in quantum information theory [BPR00, DVC00, VDDMV02, HHHH09], bounding the size of combinatorial structures like cap sets and tri-colored sum-free sets in additive combinatorics [Ede04, Tao08, ASU13, CLP17, EG17, Tao16, BCC17, KSS16, TS16], and bounding the query complexity of certain properties in algebraic property testing [KS08, BCSX10, Sha09, BX15, HX17, FK14].

There are naturally two directions in the asymptotic restriction problem, namely finding (1) constructions, i.e. matrices that carry out , and (2) obstructions, i.e. certificates that prohibit . For constructions one should think of fast matrix multiplication algorithms or efficient quantum protocols. For obstructions one should think of lower bounds in the sense of computational complexity theory. Strassen introduced in 1986 the theory of asymptotic spectra of tensors to understand the asymptotic restriction problem [Str86, Str88]. Deferring the details to the next subsection, this can be viewed as the theory of obstructions in the above sense. A remarkable result of this theory is that the asymptotic restriction problem for a family of tensors that is closed under direct sum and tensor product and contains the diagonal tensors , reduces to finding all maps that are

1. [label=()]

2. monotone under restriction

3. multiplicative under tensor product

5. normalised to have value  at the unit tensor .

Such maps are called spectral points. Here the direct sum is defined naturally as the multilinear map such that where , and the unit tensor for is defined as the multilinear map that maps to 1 if and to 0 otherwise.

Properties 1 and 2 are natural properties to obtain an obstruction. Namely, suppose is such a map, and let . If , then by definition , and 1 and 2 imply , which implies . Turning this around, if then not , so  yields an obstruction to .

Strassen in [Str91] created a family of spectral points. Let

be the set of all probability distributions on

. Strassen defined a family of maps parametrised by , named the support functionals, and he proved that the are spectral points for the family of oblique tensors, tensors whose support in some basis is an antichain, a strict (and nongeneric) subfamily of all tensors. In [Str05], for such tensors, has been given a formulation in terms of moment polytopes.

Our main result. Universal spectral points are spectral points for the family of all tensors. The construction of nontrivial universal spectral points has been an open problem for more than thirty years. We introduce maps called the quantum functionals and we prove that they are universal spectral points. The quantum functionals are defined as follows (we will carefully define the quantum concepts later). For any , define

 Fθ(t) =2Eθ(t) Eθ(t) =supgik∑i=1θ(i)H(Tri(g1,…,gk)⋅t)

where the supremum goes over invertible maps , denotes the quantum entropy i.e. von Neumann entropy, and denotes a partial trace of interpreted as a pure quantum state. To prove properties 14 we draw from the theory on invariants, quantum entropy, entanglement polytopes, Kronecker coefficients and Littlewood–Richardson coefficients. Let us briefly sketch the connection to moment polytopes, which in this context are called entanglement polytopes. Given a tensor , let be the single-system quantum marginal entropies of the tensor viewed as a quantum state. Let be the group . For a tensor , define the set , where denotes the Euclidean closure (or equivalently Zariski closure) of the orbit . It is a nontrivial fact that is a polytope, named the entanglement polytope. Then equals the following optimisation over the entanglement polytope,

 Eθ(t)=sup{k∑i=1θ(i)H(λ(i))∣∣(λ(1),…,λ(k))∈Δt},

which is a convex optimisation problem.

The definition of can in fact be extended to probability distributions on subsets of in several ways. We explore these extensions and study the properties 14. One such extension, the lower quantum functional, satisfies 1 and 4 and is super-additive and super-multiplicative. Another extension, the upper quantum functional, satisfies 1 and 4 and is sub-additive and sub-multiplicative for that are what we call noncrossing. (The Strassen support functionals in fact similarly come in an upper and a lower version.)

Asymptotic rank and asymptotic subrank. In applications (e.g. the complexity of matrix multiplication) one is often interested in asymptotic restriction to or from a unit tensor. One commonly defines the tensor rank , and the subrank , and asymptotically the asymptotic rank and the asymptotic subrank . Clearly, maps that satisfy 14 and super-multiplicativity are lower bounds on asymptotic rank, and maps that satisfy 14 and sub-multiplicativity are upper bounds on asymptotic subrank. Spectral points (maps satisfying 14) are thus between asymptotic subrank and asymptotic rank. The defining expression of asymptotic rank does not suggest any algorithm for computing its value other than computing the rank for high tensor powers. In information theory, such an expression is called a multi-letter formula, and maps satisfying 14 are called single-letter formulas.

Cap sets and slice rank. To demonstrate an application of the asymptotic spectrum we go on a brief combinatorial excursion to the cap set problem. (Full details are in Section 4.2.) A subset is called a cap set if any line in is a point, a line being a triple of points of the form . The cap set problem is to decide whether the maximal size of a cap set in grows like or like  for some when . Gijswijt and Ellenberg in [EG17], inspired by the work of Croot, Lev and Pach in [CLP17], settled this problem, showing that . Tao realised in [Tao16] that the cap set problem may naturally be phrased as the problem of computing the size of the largest main diagonal in powers of the cap set tensor where the sum is over with . Here main diagonal refers to a subset of the basis elements such that restricting to gives the tensor . A main diagonal is essentially a unit tensor, so it is sufficient to upper bound the asymptotic subrank of the cap set tensor interpreted as a tensor over . We show (in hindsight) that the cap set tensor is in the -orbit of the structure tensor of the algebra . This implies that the asymptotic spectrum of the cap set tensor and the asymptotic spectrum of coincide. The tensor is the prime example in [Str91] for which Strassen computes the whole asymptotic spectrum. The minimal point in this asymptotic spectrum, which corresponds to the asymptotic subrank, is computed by Strassen to be (see [Str91, Table 1]). This reproves the bound by Ellenberg–Gijswijt.

Although our universal spectral points, the quantum functionals, do not play a role for cap sets (we cannot work over for this problem), we think that a better understanding of our universal spectral points construction may lead to further progress on related combinatorial questions.

In the study of the cap set problem, slice rank and multi-slice rank were introduced as upper bounds on subrank [Tao16, Nas17]. We show that asymptotically the (upper) quantum functionals and the Strassen (upper) support functionals are an upper bound on slice rank and multi-slice rank. As a consequence we prove that for the family of tight 3-tensors asymptotic subrank and asymptotic slice rank coincide. For complex tensors we characterise the asymptotic slice rank in terms of the quantum functionals.

### 1.2 Asymptotic spectra for tensors

To understand why we focus on maps satisfying the properties 14 we have to give a brief introduction to Strassen’s theory of asymptotic spectra for tensors.

We begin by putting an equivalence relation on tensors to get rid of trivialities. Restriction and asymptotic restriction are both preorders (reflexive and transitive). We say is isomorphic to , and write , if there are bijective linear maps such that . We say and are equivalent, and write , if there are null maps and such that . The equivalence relation  is in fact the equivalence relation generated by the restriction preorder . Let be the set of -equivalence classes of -multilinear maps of order . Direct sum and tensor product naturally carry over to , and becomes a semiring with additive unit and multiplicative unit (more precisely, the equivalence classes of those tensors). Restriction induces a partial order on (reflexive, antisymmetric, transitive), and asymptotic restriction  induces a preorder on . Both behave well with respect to the semiring operations, and naturally if and only if .

The theory of asymptotic spectra revolves around the following theorem proved by Strassen [Str86, Str88]. Given a topological space we denote by  the semiring of continuous maps .

###### Theorem 1.1 (Spectral theorem).

For any semiring there are

1. a compact space

2. a homomorphism of semirings

such that separates points and such that if and only if pointwise on , and the pair is essentially unique. We call an asymptotic spectrum for . Explicitly, may be taken as follows:

1. .

2. where .

We call this pair the asymptotic spectrum of and refer to it as .

Theorem 1.1 is proved by a nontrivial reduction to the structure theory of Stone–Kadison–Dubois for a certain class of ordered rings (see [BS83]). We will not go into the details of this proof here, nor do we elaborate on how is unique. We have taken the name spectral theorem from the talk [Str12].

###### Remark 1.2.

We note that may equivalently be defined with degeneration  instead of restriction . Over , we say degenerates to , written, , if is in the Euclidean closure (or equivalently Zariski closure) of the orbit . It is a nontrivial fact from algebraic geometry (see [Kra84, Lemma III.2.3.1] or [BCS97]) that there is a degeneration if and only if there are linear maps such that for some elements , where is the field extended with the formal variable . The latter definition of degeneration is valid when is replaced by an arbitrary field and that is how degeneration is defined for an arbitrary field. Degeneration is weaker than restriction: implies . Asymptotically, however, the notions coincide: if and only if . We mention that, analogous to restriction, degeneration gives rise to border rank and border subrank, , .

###### Remark 1.3.

Let be a family of tensors over . Let be a field extension. We may view as a family of tensors over . The asymptotic spectra of and are equal [Str88, Theorem 3.10]. In particular, if , then can be viewed as a tensor in over any field , and by the above, if is the prime subfield of , then , and e.g. . A well-known example is that the exponent of matrix multiplication over depends only on the characteristic of .

We reduced the asymptotic restriction problem to the problem of computing the asymptotic spectrum of or a subsemiring . An element is called a spectral point. It is clear that constructing explicit elements in the asymptotic spectrum  of all tensors is the holy grail in the development of the theory of asymptotic spectra of tensors. Elements of  are called universal spectral points, and they correspond precisely to maps satisfying , , , and whenever  (i.e. properties 14).

###### Example 1.4 (Gauge points).

In [Str88, Equation 3.10] universal spectral points are given. Namely, given a multilinear map , let and similarly define for . Define . The maps are universal spectral points and are named gauge points.

In some applications it is useful to think of the asymptotic spectrum as a compact subspace of .

###### Definition 1.5.

Let be a set. Let be the semiring generated by . A homomorphism is determined by the values . We may thus identify the asymptotic spectrum of the semiring generated by with a subspace of .

1. .

2. .

We call the asymptotic spectrum of , and write . When we will denote by .

We finish by stating the important relationship between the asymptotic spectrum and the asymptotic (sub)rank, which follows from Theorem 1.1 (see [Str88, Theorem 3.8]).

###### Proposition 1.6.

Let be a semiring. Let be an asymptotic spectrum of . Let . Then

 \lx@converttounder\wtildeQ(a)=minξ∈Δϕ(a)(ξ)\lx@converttounder\wtildeR(a)=maxξ∈Δϕ(a)(ξ).
###### Remark 1.7.

Obviously and are -monotones and have value on . They are not universal spectral points however. Namely, the asymptotic rank of , and is 2, whereas the tensor product equals the matrix multiplication tensor whose asymptotic rank is strictly smaller than . With the same tensors one shows that asymptotic subrank is not multiplicative.

### 1.3 This paper

Main results. The main results of this paper are summarised as follows.

1. We construct for the first time a nontrivial family of maps

 Fθ:{k-tensors}→R≥0,

named quantum functionals, that are nonincreasing under restriction , normalised on the unit tensor , additive under direct sum and multiplicative under tensor product , i.e. universal spectral points, advancing Strassen’s theory of asymptotic spectra.

2. We connect Strassen’s asymptotic spectra to entanglement polytopes and the quantum marginal problem. This is to our knowledge the first time that information about asymptotic transformations is obtained from entanglement polytopes, as opposed to information about single-copy transformations.

3. We put recent progress on the cap set problem in the framework of Strassen, which may prove useful in solving variations on the cap set problem. We characterise asymptotic slice rank in terms of the quantum functionals and show that asymptotic slice rank coincides with asymptotic subrank for tight 3-tensors.

Paper overview. In Section 2 we set the scene by reviewing the theory of the Strassen upper and lower support functionals  and  introduced in [Str91]. This is important background material for our work. The support functionals allow us to compute elements in the asymptotic spectrum for a class of tensors called oblique tensors. Our exposition follows closely the exposition in [Str91] with the exception that we consider multilinear maps with instead of bilinear maps . Strassen already observed that his results generalise to the multilinear regime, but we will make this explicit for the sake of understanding and comparison to our results.

Section 3 contains our main result. Here we introduce two new families of functionals over the complex numbers. These functionals are denoted by and  and called the upper and lower quantum functionals. Both functionals are -monotone. The upper quantum functional is sub-additive and sub-multiplicative when is what we call noncrossing. The lower quantum functional  is super-additive and super-multiplicative. For singleton the two functionals coincide and thus yield an additive, multiplicative -monotone, a universal spectral point.

In Section 4 we consider several families of tensors. We introduce the family of free tensors. We show that for free tensors, the Strassen upper support functional coincides with the quantum functionals for singleton . This is useful computationally, since the upper support functional is defined as a minimisation while the quantum functionals are defined as a maximisation. We next compute generic values of the quantum functionals for tensor formats in which a quantum state with completely mixed marginals exists. Our results extend over the results by Verena Tobler [Tob91] on the generic value of the upper support functional. Finally, we reprove recent results on the cap set problem, by reducing this problem to a result of Strassen on reduced polynomial multiplication.

In Section 5 we show that asymptotically the Strassen upper support functional and the upper quantum functional are upper bounds for slice rank and multi-slice rank, recently introduced -monotones that are neither super-multiplicative nor sub-multiplicative. As a consequence, we find that slice rank coincides asymptotically with subrank for tight 3-tensors. We additionally show that the quantum functionals characterise asymptotic slice rank for complex tensors.

## 2 Strassen support functionals

In this section the field is arbitrary. After having introduced the concept of the asymptotic spectrum of tensors in [Str88], Strassen constructed a nontrivial family of spectral points in the asymptotic spectrum of oblique tensors in [Str91]. Oblique tensors are tensors for which the support is an antichain in some basis. In plain English, Strassen constructed a family of functions from -tensors to  that are -monotone, are normalised to attain value at , and, when restricted to oblique tensors, are additive under and multiplicative under . He calls his functions the support functionals, because the support of a tensor plays an important role in the definition. Not all tensors are oblique, and obliqueness is not a generic property. Many tensors that are of interest in the field of algebraic complexity theory, however, turn out to be oblique, notably the structure tensor of the algebra of matrices. This section is devoted to explaining the construction of these spectral points for oblique tensors. We do this not only to set the scene and provide benchmarks for our new functionals, but also because the Strassen support functionals remain relevant today, for example in the context of combinatorial problems like the cap set problem, as we will make clear in Section 4.2.2.

The construction goes in four steps. First we define for probability distributions on the upper support functional , which is -monotone, normalised at , sub-additive under and sub-multiplicative under . Second we define the lower support functional , which is -monotone, normalised at , super-additive under and super-multiplicative under . Third we show that for any -tensor . Fourth we show that for oblique tensors the upper support functional and lower support functional coincide, which gives a spectral point.

Notation. We use the following standard notation. For any natural number  we write  for the set . For any finite set , let be the set of all probability distributions on . For any probability distribution the Shannon entropy of is defined as with understood as 0. The support of is . Given finite sets and a probability distribution on the product set we denote the marginal distribution of on  by , that is, for any .

We set some specific notation for this text. Let . For , let

be a vector space over

of dimension . Let be a finite set. Let be a multilinear map. We order the set naturally by . When the sets are ordered, we give the product order, defined by iff for all : .

### 2.1 Upper support functional

We begin with introducing the upper support functional. We naturally define the notion of the support of a multilinear map .

###### Definition 2.1.

Let be a multilinear map. Let  denote the set of -tuples of bases for . That is, each element is a -tuple in which is a basis of . Define the support of with respect to as

 suppCf\coloneqq{(α1,…,αk)∈[n1]×⋯×[nk]∣∣f(v1,α1,…,vk,αk)≠0}.
###### Definition 2.2.

Let . Let be a probability distribution. Define as the -weighted average of the Shannon entropies of the marginal distributions  of ,

 Hθ(P)=∑i∈[k]θ(i)H(Pi).

Let be a nonempty subset. Define  as the maximum  over all probability distributions ,

 Hθ(Φ)=maxP∈P(Φ)Hθ(P). (1)
###### Definition 2.3 (Strassen upper support functional).

Let . Let be a nonzero multilinear map. Define by

 ρθ(f) =minC∈C(f)Hθ(suppCf) and define ζθ by ζθ(f) =2ρθ(f).

We call the Strassen upper support functional and we call the logarithmic Strassen upper support functional. When is 0 we naturally define and .

The key properties of the upper support functional are as follows.

###### Theorem 2.4 ([Str91]).

Let and be multilinear maps. Let . The Strassen upper support functional has the following properties.

1. for .

2. .

3. .

4. If , then .

5. .

(Note that is not just sub-additive, but even additive.) One verifies directly that statement 1 and 5 of Theorem 2.4 are true. Statement 2–4 can be proved with generalisations of the arguments in [Str91, Section 2].

In the rest of this section we discuss an important alternative characterisation of the upper support functional in terms of filtrations. This alternative definition we need later to relate the upper and lower support functionals.

###### Definition 2.5.

A proper filtration or complete flag of a vector space is a sequence of subspaces of with and for . (In this text all flags will be decreasing.) We denote by the set of -tuples of complete flags of . For we define the support of with respect to as

 suppFf={(α1,…,αk)∣f(V1,α1×⋯×Vk,αk)≠0}⊆[n1]×⋯×[nk].

The alternative characterisation of the upper support functional in terms of filtrations is as follows. We refer to [Str91, Section 2] for the proof.

.

###### Remark 2.7.

The characterisation of as a minimisation over filtrations can be thought of as an asymptotic version of the filtration method explained in [BCS97, Exercise 15.13].

### 2.2 Lower support functional

The upper support functional has a companion called the lower support functional which we introduce here. We shift focus from the support to the maximal points in the support.

###### Definition 2.8.

Let be a subset. We define the maximal points of with respect to the product order of the natural orders on as

###### Definition 2.9.

Let be a nonzero multilinear map. Given we set the notation . Define by

 ρθ(f)=maxC∈C(f)Hθ(MCf)

and define by

 ζθ(f)=2ρθ(f).

We call the Strassen lower support functional and we call the logarithmic Strassen lower support functional. When is 0 we naturally define and .

###### Theorem 2.10 ([Str91]).

Let and . Let . The Strassen lower support functional has the following properties.

1. for .

2. .

3. .

4. If , then .

5. .

###### Remark 2.11.

Regarding statement 2 in Theorem 2.10, Bürgisser [Bü90] shows that the lower support functional is not in general additive under the direct sum when for all . See also [Str91, Comment (iii)]. In particular, this implies that the upper support functional and the lower support functional  are not equal in general, the upper support functional being additive. In fact, to show that the lower support functional is not additive, Bürgisser first shows that (with algebraically closed) the typical value of on equals ; on the other hand, Tobler [Tob91] shows that the typical value of on equals . (So even generically and are different on .) We discuss typical values in Section 4.4.

One verifies directly that statement 1 and 5 of Theorem 2.10 are true. For the proofs of statements 2–4 we refer to [Str91, Section 3].

We finish this section by giving an alternative definition of in terms of complete flags, in the same spirit as Proposition 2.6 for the upper support functional. Let us first make some general remarks about how (the maximal points in) the support with respect to a basis and the support with respect to a flag are related.

###### Definition 2.12.

Let be a subset. We define the downward closure of with respect to the product of the natural orders on as

To any we can naturally associate a -tuple of complete flags , by defining the subspace as .

The following key properties of the downward closure and the maximal points follow directly from the definitions.

###### Lemma 2.13.

Let and be associated. Then

 =suppFf, maxsuppCf =maxsuppFf.

For we have set the notation . For we similarly set the notation .

for .

###### Proof.

Let and be associated. Then by Lemma 2.13 and hence . Since any is associated to some and vice versa we conclude that

 maxC∈C(f)Hθ(MCf)=maxF∈Fc(f)Hθ(MFf).\qed

### 2.3 Comparing the support functionals

Strassen calls his functionals upper and lower because he proves that the upper support functional is at least the lower support functional [Str91, Corollary 4.3].

###### Theorem 2.15.

.

This is a useful property when doing computations, since is defined as a minimisation and is defined as a maximisation.

In fact, in [Str91, Corollary 4.2] Strassen first proves and then proves Theorem 2.15 as a corollary. The purpose of this section is to give a more direct proof of Theorem 2.15 for the benefit of the reader. The proof essentially comes down to the following proposition.

###### Proposition 2.16.

Let . Let . There are permutations such that .

###### Proof.

Let be some -dimensional vector space and let and be complete flags of such that

 Vα =Span{vα,vα+1,…} Wβ =Span{wβ,wβ+1,…}.

Define the map

 ϕ:[n]→[n]:β↦max{α∈[n]:Vα∩(wβ+Wβ+1)≠∅}. (2)

The map is injective. Let with and suppose . Then (2) gives

 (F×vα+Vα+1)∩(wβ+Wβ+1) ≠∅ (3) Vα+1∩(wβ+Wβ+1) =∅ (4) (F×vα+Vα+1)∩(wγ+Wγ+1) ≠∅ (5)

If , then using (3) and (5) we may obtain a contradiction to (4). We conclude that .

We turn to and . Write

 F =((V1,α1),…,(Vk,αk)) G =((W1,β1),…,(Wk,βk))

and for each pair of complete flags of we define the permutation in the same way as above. We prove . Let be in . Let . Then by definition of the the intersection is not empty. Choose

 ¯¯¯¯¯¯¯¯¯¯wi,βi∈Vi,αi∩(wi,βi+Wi,βi+1).

Since is multilinear we have for some

 f(¯¯¯¯¯¯¯¯¯¯¯w1,β1,…,¯¯¯¯¯¯¯¯¯¯¯wk,βk)=f(w1,β1,…,wk,βk)+∑β′>βxβ′f(w1,β′1,…,wk,β′k)

with the sum over tuples that are strictly larger than in the product order. Since is a maximal element in with respect to the product order, the sum over equals zero. We conclude that . Therefore, is not zero and thus . ∎

The above proof of Proposition 2.16 may more naturally be phrased in the language of Schubert cells, when , as follows. We use the notation and definitions of [Bri05] with the difference that our complete flags are decreasing instead of increasing so we need to use the group of lower unitriangular matrices  instead of the group of upper unitriangular matrices .

###### Proof.

Let be the unipotent subgroup of lower triangular matrices with ones on the diagonal. Let be the variety of complete (decreasing) flags in . The group naturally acts on . Let the symmetric group  act on via its natural action on the standard basis of . It is well-known (see e.g. [Bri05, Proposition 1.2.1]) that the variety is the disjoint union of the -orbits over all permutations  in the symmetric group , where  is the standard flag