Weak Thin Schubert Cell in Polymatroids

• Weak thin Schubert cell is defined as the moduli functor of weak F-representations with fixed support by imposing only 3-term Plücker relations, and it is represented by Spec P_J.
• It establishes universal properties by capturing key algebraic structures that differentiate it from strong Schubert cells while embedding into toric spaces.
• This construction supports functoriality, duality, and torus actions, promoting decompositions over idempotent tracts and providing insights for tropical geometry applications.

Searching arXiv for the cited paper to ground the article in the source. In the representation theory of discrete polymatroids over tracts, a weak thin Schubert cell is the moduli functor obtained by imposing only the $3$-term Plücker relations on Plücker-coordinate data with fixed support $J\subseteq \Delta^n_r$. For a polymatroid $J$, it assigns to a tract $F$ the quotient of all weak $F$-representations of $J$ by scalar rescaling, and it is canonically isomorphic to $\operatorname{Spec} P_J$, where $P_J$ is the universal object cut out by those $3$-term relations. Within the same framework, the strong thin Schubert cell is defined by all Plücker relations, while passage to torus orbits yields the realization space represented by the foundation $F_J$. The theory places weak thin Schubert cells among universal tracts, universal pastures, cross-ratio coordinates, torus embeddings, and idempotent decompositions (Baker et al., 19 Jul 2025).

1. Definition via $J\subseteq \Delta^n_r$0-term Plücker relations

Let $J\subseteq \Delta^n_r$1 be positive integers and let

$J\subseteq \Delta^n_r$2

be an $J\subseteq \Delta^n_r$3-convex set (“polymatroid”) of effective rank

$J\subseteq \Delta^n_r$4

where $J\subseteq \Delta^n_r$5 and $J\subseteq \Delta^n_r$6 (Baker et al., 19 Jul 2025).

For a tract $J\subseteq \Delta^n_r$7, a weak $J\subseteq \Delta^n_r$8-representation of $J\subseteq \Delta^n_r$9 is a function

$J$0

whose support

$J$1

equals $J$2, and which satisfies the $J$3-term Plücker relations. Explicitly, for every $J$4 and every quadruple of distinct indices $J$5 in $J$6 such that

$J$7

all lie in $J$8, one requires

$J$9

where $F$0 is the null-set of $F$1. Equivalently, writing

$F$2

for that sum, the condition is

$F$3

whenever all six corresponding points lie in $F$4 (Baker et al., 19 Jul 2025).

The weak thin Schubert cell is then the functor

$F$5

Its defining feature is that the support is fixed exactly to $F$6, while only the $F$7-term Plücker constraints are imposed. This distinguishes it from the strong thin Schubert cell, where all Plücker relations of arbitrary size are imposed.

2. Universal object and scheme-theoretic realization

To represent the weak thin Schubert cell, one introduces one coordinate $F$8 for each $F$9, considers the free tract over $F$0 on these symbols, and imposes precisely the null-relations given by the $F$1-term Plücker sums (Baker et al., 19 Jul 2025). Concretely,

$F$2

where $F$3 denotes the free tract (or pasture) construction and $F$4 the ideal generated by the $F$5-term relations for all choices of $F$6.

The universal property is

$F$7

Equivalently, viewing $F$8 as an affine “band-scheme,” one has

$F$9

This representability statement makes the weak thin Schubert cell an algebraic object rather than merely a set-valued construction. In the terminology of the abstract, restricting to the $J$0-term Plücker relations yields the weak thin Schubert cell, and these are represented by the universal pasture; in the detailed construction, the representing object is denoted $J$1 (Baker et al., 19 Jul 2025).

3. Relation to the universal tract and the foundation

The weak thin Schubert cell sits between two related universal constructions: the universal tract $J$2, obtained by imposing all Plücker relations, and the foundation $J$3, obtained after quotienting by diagonal torus action and passing to multidegree-zero data (Baker et al., 19 Jul 2025).

Object	Defining relations / quotient	Represents
$J$4	all Plücker relations of arbitrary size $J$5	strong thin Schubert cell
$J$6	only the $J$7-term Plücker relations	weak thin Schubert cell
$J$8	multidegree-zero part of $J$9	realization space

The universal tract $\operatorname{Spec} P_J$0 is defined analogously to $\operatorname{Spec} P_J$1, but with all Plücker relations of arbitrary size $\operatorname{Spec} P_J$2. The natural map

$\operatorname{Spec} P_J$3

is bijective on underlying sets. Consequently, the strong and weak thin Schubert cells have the same $\operatorname{Spec} P_J$4-points whenever $\operatorname{Spec} P_J$5 is an “excellent” tract.

The foundation $\operatorname{Spec} P_J$6 is obtained by modding out by the diagonal action of the torus $\operatorname{Spec} P_J$7, i.e. by passing from Plücker coordinates to cross-ratio coordinates. Concretely,

$\operatorname{Spec} P_J$8

where

$\operatorname{Spec} P_J$9

is the multidegree recording which $P_J$0 appear. The corresponding realization space is

$P_J$1

and it satisfies

$P_J$2

The abstract further states that the foundation of a polymatroid is generated by cross ratios, and that a possibly incomplete list of multiplicative relations between cross ratios is described. A common source of confusion is to identify the weak thin Schubert cell directly with the realization space; the theory separates these objects by an explicit torus quotient.

4. Torus embedding, degeneracy locus, and orbit structure

Every weak representation

$P_J$3

with $P_J$4 defines a point in the torus $P_J$5 by its nonzero Plücker coordinates. After dividing by $P_J$6, one obtains the Plücker embedding

$P_J$7

(Baker et al., 19 Jul 2025).

Within this toric ambient space, the paper identifies a subgroup by degenerate $P_J$8-term relations. Requiring only the degenerate $P_J$9-term relations suffices to cut out the image of the strong representation space

$3$0

this subgroup is called the degeneracy locus $3$1. The resulting inclusions are

$3$2

There is also a natural torus action. The torus

$3$3

acts on $3$4 by

$3$5

Its orbits are exactly the fibers of the map to the realization space $3$6. In the language of the abstract, passing to torus orbits yields the realization space. This suggests that the weak thin Schubert cell retains both intrinsic realization data and additional toric degrees of freedom encoded in the Plücker coordinates.

5. Idempotent tracts and product decomposition

A particularly rigid description appears over idempotent tracts. Assume $3$7 is idempotent, meaning

$3$8

Write

$3$9

for the unique decomposition into indecomposable components (Baker et al., 19 Jul 2025).

Then the weak thin Schubert cell decomposes functorially as

$F_J$0

Equivalently, the stabilizer of a point under $F_J$1 has codimension $F_J$2, so each $F_J$3-orbit is isomorphic to

$F_J$4

In particular, the lineality space of the weak cell, i.e. the orbit of the trivial representation, is a distinguished torus factor $F_J$5.

The abstract states this in equivalent geometric terms: over idempotent tracts, thin Schubert cells contain a canonical torus orbit and split naturally as a product of the realization space with this distinguished torus. A plausible implication is that, in the idempotent setting, the realization-theoretic content is separated cleanly from the toric lineality directions.

6. Functoriality, duality, direct sums, and examples

The weak thin Schubert cell has several structural compatibilities (Baker et al., 19 Jul 2025). If

$F_J$6

is any polymatroid embedding—specifically a minor, translation, coordinate inclusion, or permutation—then it induces compatible maps on weak thin Schubert cells and on realization spaces via pull-back

$F_J$7

Duality is canonical at the level of universal objects: $F_J$8 covering the involution $F_J$9. Hence

$J\subseteq \Delta^n_r$00

and likewise for realization spaces.

For direct sums, if

$J\subseteq \Delta^n_r$01

then

$J\subseteq \Delta^n_r$02

The examples highlight the scope of the notion. For the tropical hyperfield $J\subseteq \Delta^n_r$03, one has weak $J\subseteq \Delta^n_r$04 strong, and $J\subseteq \Delta^n_r$05 is the local Polydressian stratum of $J\subseteq \Delta^n_r$06-convex support $J\subseteq \Delta^n_r$07, whose logarithmic image is the “polymatroid Dressian” $J\subseteq \Delta^n_r$08 modulo constants. In rank $J\subseteq \Delta^n_r$09, $J\subseteq \Delta^n_r$10, one recovers Knutson–Tao hives and the saturation theorems.

For small polymatroids, explicit foundations can be computed. For

$J\subseteq \Delta^n_r$11

one finds

$J\subseteq \Delta^n_r$12

so that

$J\subseteq \Delta^n_r$13

For

$J\subseteq \Delta^n_r$14

one computes

$J\subseteq \Delta^n_r$15

These constructions place the weak thin Schubert cell as a central intermediary between Plücker-coordinate representation spaces and cross-ratio realization spaces: it records weak tract-valued representations with fixed support, embeds canonically into a projective torus, and interacts compatibly with minors, duality, direct sums, and idempotent splitting.