Lower-Upper Varieties: Schemes, Lattices, Nef Cones

• Lower-Upper Varieties are structures defined by simultaneous lower and upper constraints on objects in geometry and combinatorics, as seen in matrix schemes and lattice theory.
• Matrix-geometric lower-upper schemes enforce triangular conditions on XY and YX, yielding irreducible components indexed by partial permutations and computed via combinatorial pipe dreams.
• In lattice theory and toric geometry, the lower-upper framework categorizes modular varieties and provides explicit polyhedral bounds for nef cones through toric embeddings.

“Lower-Upper Varieties” is not a single standardized notion. In the literature, the phrase and its close variants occur in several distinct settings united by a common pattern: a geometric, combinatorial, or lattice-theoretic object is constrained simultaneously by a “lower” condition and an “upper” condition. The most literal usage is the matrix-geometric one, where the lower-upper scheme

$$E=\{(X,Y): XY \text{ lower triangular},\ YX \text{ upper triangular}\}$$

has irreducible components $E_\pi$ called lower-upper varieties (Knutson et al., 2 Sep 2025). In lattice theory, the phrase naturally refers to varieties that are both lower-modular and upper-modular, a class completely determined in the lattice $\mathbf{EPI}$ of epigroup varieties (Shaprynskii et al., 2014). In birational and toric geometry, one also encounters a “lower-upper” structure on invariants such as nef cones, where explicit polyhedral lower bounds and upper bounds are extracted from toric embeddings (Gibney et al., 2010).

1. Terminological scope

In the available literature, “lower-upper” has at least three mathematically precise uses.

Setting	Object	Characteristic statement
Matrix geometry	Lower-upper varieties $E_\pi$	$XY$ lower triangular and $YX$ upper triangular
Lattice theory	Varieties both lower-modular and upper-modular	In $\mathbf{EPI}$, these are exactly four neutral varieties
Nef-cone approximation	Lower-upper structure on $\Nef(Y)$	Polyhedral lower and upper bounds from a toric embedding

The matrix-geometric usage is the most literal. For $m,n\in\mathbb N$, the rectangular lower-upper scheme is

$E := \left\{ (X,Y) \in \Mat(m,n,\mathbb C)\times \Mat(n,m,\mathbb C):\ X Y\text{ lower triangular, } Y X\text{ upper triangular}\right\},$

and its irreducible components $E_\pi$0, indexed by partial permutations $E_\pi$1, are the lower-upper varieties in the strict geometric sense (Knutson et al., 2 Sep 2025). In the square case $E_\pi$2, later work recalls that this scheme was introduced in $E_\pi$3, while the rectangular case is needed for induction and for the hybrid-pipe-dream formalism (Knutson et al., 2 Sep 2025).

The lattice-theoretic usage is different. There, “lower-upper varieties” are varieties that are simultaneously lower-modular and upper-modular as elements of a lattice of varieties. In $E_\pi$4, the interaction is especially rigid: an epigroup variety is neutral if and only if it is both lower-modular and upper-modular (Shaprynskii et al., 2014). A broader survey shows analogous questions for $E_\pi$5, $E_\pi$6, and related sublattices, with complete results in some cases and open problems in others (Vernikov, 2013).

A third usage is best read as a “lower-upper structure” attached to a variety rather than a new class of varieties. Given a projective variety $E_\pi$7 embedded in a toric variety $E_\pi$8, explicit cones

$E_\pi$9

provide polyhedral lower bounds and upper bounds for $\mathbf{EPI}$0, yielding a systematic lower-upper approximation framework (Gibney et al., 2010).

2. Lower-upper schemes and their irreducible components

The geometric theory begins with the scheme

$\mathbf{EPI}$1

whose defining equations are

$\mathbf{EPI}$2

Thus $\mathbf{EPI}$3 is forced to lie in the lower-triangular sector and $\mathbf{EPI}$4 in the upper-triangular sector (Knutson et al., 2 Sep 2025). The square case $\mathbf{EPI}$5 was introduced earlier, but the rectangular generalization is structurally important because many inductive arguments pass through it (Knutson et al., 2 Sep 2025).

The irreducible components are indexed by maximal-rank $\mathbf{EPI}$6 partial permutation matrices, equivalently by injective maps $\mathbf{EPI}$7. For such $\mathbf{EPI}$8,

$\mathbf{EPI}$9

and equivalently

$E_\pi$0

Here $E_\pi$1 and $E_\pi$2 are the invertible lower- and upper-triangular groups of the appropriate sizes (Knutson et al., 2 Sep 2025). Geometrically, $E_\pi$3 is the closure of the locus where the nonzero diagonal entries of $E_\pi$4 and $E_\pi$5 are matched by $E_\pi$6.

A central reformulation is the flux equation

$E_\pi$7

This identifies the diagonal entry $E_\pi$8 with $E_\pi$9 and is the algebraic origin of the pipe-dream connectivity data (Knutson et al., 2 Sep 2025). In the square case, the same components are indexed by permutations $XY$0, and one has

$XY$1

as well as

$XY$2

which makes the $XY$3-orbit geometry completely explicit (Knutson et al., 2024).

The distinction between the global scheme $XY$4 and its irreducible varieties $XY$5 is essential. The scheme structure matters in degeneration arguments, and later work repeatedly notes that flat limits may have embedded components, even though those lower-dimensional embedded pieces do not affect the top-dimensional equivariant class formula (Knutson et al., 2 Sep 2025).

3. Pipe dreams, equivariant classes, and degeneration theory

The modern structure theory of lower-upper varieties is combinatorial. In the square case, generic pipe dreams produce a polynomial

$XY$6

and the fundamental class formula is

$XY$7

Equivalently, the $XY$8-equivariant cohomology class of $XY$9 is computed by a sum over generic pipe dreams with tile weights $YX$0, $YX$1, and $YX$2, according to tile type (Knutson et al., 2024).

The rectangular theory sharpens this. For each hybridization $YX$3, hybrid generic pipe dreams yield a polynomial $YX$4, and Theorem $YX$5 shows that it is independent of $YX$6, so the notation becomes $YX$7. The central class formula is

$YX$8

Here the torus weights are

$YX$9

so $\mathbf{EPI}$0 lies in

$\mathbf{EPI}$1

The same work proves a divided-difference type recurrence for both $\mathbf{EPI}$2 and $\mathbf{EPI}$3, thereby identifying the combinatorial and geometric quantities exactly (Knutson et al., 2 Sep 2025).

The degeneration theorem explains why the combinatorics is not ad hoc. For each hybridization $\mathbf{EPI}$4, there is a degeneration of $\mathbf{EPI}$5 to a union of complete intersections $\mathbf{EPI}$6, indexed by hybrid generic pipe dreams $\mathbf{EPI}$7, and the contribution of $\mathbf{EPI}$8 to $\mathbf{EPI}$9 is $\Nef(Y)$0 (Knutson et al., 2 Sep 2025). In the square case, an analogous degeneration yields a union of quadratic complete intersections $\Nef(Y)$1, possibly together with lower-dimensional embedded components, and the class of each $\Nef(Y)$2 is the corresponding summand in $\Nef(Y)$3 (Knutson et al., 2024).

One of the conceptual innovations is the flux formalism. Fluxes $\Nef(Y)$4 are attached to edges of the matrix grid, and local equalities among these fluxes reconstruct the pipe-dream conditions. In the rectangular theory, pipe dreams are derived from equalities among edge fluxes rather than postulated independently; in the square theory, the defining ideal

$\Nef(Y)$5

degenerates in a way that preserves flux equations and yields local relations such as

$\Nef(Y)$6

forcing the admissible tile patterns (Knutson et al., 2 Sep 2025, Knutson et al., 2024).

These formulas place lower-upper varieties between matrix Schubert geometry and commuting-variety geometry. The projection $\Nef(Y)$7 maps $\Nef(Y)$8 onto the rectangular matrix Schubert variety $\Nef(Y)$9 (Knutson et al., 2 Sep 2025), while in the square case $m,n\in\mathbb N$0 for the commuting variety $m,n\in\mathbb N$1, and

$m,n\in\mathbb N$2

gives a concrete degree formula (Knutson et al., 2024). The $m,n\in\mathbb N$3 or $m,n\in\mathbb N$4 limits recover classic and bumpless pipe dream formulas for double Schubert polynomials, showing that lower-upper geometry is strictly richer than either specialization (Knutson et al., 2 Sep 2025, Knutson et al., 2024).

4. Lower-upper varieties in lattices of algebraic varieties

In lattice theory, “lower-upper varieties” refers to varieties that are both lower-modular and upper-modular as elements of a lattice of varieties. The cleanest complete result is in $m,n\in\mathbb N$5, the lattice of all epigroup varieties. There, Theorem 1.1 states that for an epigroup variety $m,n\in\mathbb N$6, the following are equivalent: $m,n\in\mathbb N$7 is neutral; $m,n\in\mathbb N$8 is costandard; $m,n\in\mathbb N$9 is simultaneously lower-modular and upper-modular; and

$E := \left\{ (X,Y) \in \Mat(m,n,\mathbb C)\times \Mat(n,m,\mathbb C):\ X Y\text{ lower triangular, } Y X\text{ upper triangular}\right\},$0

Thus the lower-upper epigroup varieties are exactly four neutral varieties (Shaprynskii et al., 2014).

This theorem is rigid because the separate lower-modular and upper-modular classifications are much larger. Lower-modular epigroup varieties are completely classified by

$E := \left\{ (X,Y) \in \Mat(m,n,\mathbb C)\times \Mat(n,m,\mathbb C):\ X Y\text{ lower triangular, } Y X\text{ upper triangular}\right\},$1

with $E := \left\{ (X,Y) \in \Mat(m,n,\mathbb C)\times \Mat(n,m,\mathbb C):\ X Y\text{ lower triangular, } Y X\text{ upper triangular}\right\},$2 and $E := \left\{ (X,Y) \in \Mat(m,n,\mathbb C)\times \Mat(n,m,\mathbb C):\ X Y\text{ lower triangular, } Y X\text{ upper triangular}\right\},$3 $E := \left\{ (X,Y) \in \Mat(m,n,\mathbb C)\times \Mat(n,m,\mathbb C):\ X Y\text{ lower triangular, } Y X\text{ upper triangular}\right\},$4-reduced (Shaprynskii et al., 2014). By contrast, upper-modular epigroup varieties are completely classified only in the strongly permutative case: either

$E := \left\{ (X,Y) \in \Mat(m,n,\mathbb C)\times \Mat(n,m,\mathbb C):\ X Y\text{ lower triangular, } Y X\text{ upper triangular}\right\},$5

with $E := \left\{ (X,Y) \in \Mat(m,n,\mathbb C)\times \Mat(n,m,\mathbb C):\ X Y\text{ lower triangular, } Y X\text{ upper triangular}\right\},$6 and $E := \left\{ (X,Y) \in \Mat(m,n,\mathbb C)\times \Mat(n,m,\mathbb C):\ X Y\text{ lower triangular, } Y X\text{ upper triangular}\right\},$7 commutative satisfying

$E := \left\{ (X,Y) \in \Mat(m,n,\mathbb C)\times \Mat(n,m,\mathbb C):\ X Y\text{ lower triangular, } Y X\text{ upper triangular}\right\},$8

or

$E := \left\{ (X,Y) \in \Mat(m,n,\mathbb C)\times \Mat(n,m,\mathbb C):\ X Y\text{ lower triangular, } Y X\text{ upper triangular}\right\},$9

with $E_\pi$00 Abelian and $E_\pi$01 commutative satisfying

$E_\pi$02

The simultaneous lower-upper condition collapses this much larger landscape to four points (Shaprynskii et al., 2014).

The semigroup-variety survey shows that analogous phenomena occur in $E_\pi$03 and $E_\pi$04, but with different final lists. In $E_\pi$05, a semigroup variety is both lower-modular and upper-modular if and only if it is both distributive and codistributive, equivalently costandard, equivalently neutral, and this happens exactly for

$E_\pi$06

In $E_\pi$07, simultaneous COM-lower-modularity and COM-upper-modularity is equivalent to COM-neutrality and is characterized by $E_\pi$08 or by varieties of the form $E_\pi$09 with $E_\pi$10 and $E_\pi$11 (Vernikov, 2013).

For commutative semigroup varieties, the “upper” side is itself unusually rigid. A commutative semigroup variety $E_\pi$12 is upper-modular if and only if it is codistributive, and this happens exactly in one of three cases: $E_\pi$13 or

$E_\pi$14

with $E_\pi$15 satisfying

$E_\pi$16

or

$E_\pi$17

with $E_\pi$18 an Abelian periodic group variety, $E_\pi$19, and $E_\pi$20. Moreover, costandard is exactly the modular plus upper-modular case (Vernikov, 2015).

A common misconception is that lower-modular and upper-modular classifications are known in equal generality. The literature says otherwise. Lower-modular elements are completely classified in several settings, including $E_\pi$21, $E_\pi$22, and $E_\pi$23, while upper-modular classifications are often complete only under additional hypotheses such as strong permutativity or commutativity (Shaprynskii et al., 2014, Vernikov, 2013, Vernikov, 2015).

5. Lower-upper structures on nef cones

A different but conceptually related use of “lower-upper” appears in the study of nef cones. Given an embedding

$E_\pi$24

of a $E_\pi$25-dimensional projective variety into an $E_\pi$26-dimensional normal toric variety, one constructs three cones in $E_\pi$27: $E_\pi$28 together with

$E_\pi$29

Pulling back via $E_\pi$30, one obtains explicit lower and upper bounds for $E_\pi$31 (Gibney et al., 2010).

The lower cones have distinct geometric meanings. The cone

$E_\pi$32

is the strongest lower bound and coincides with the toric nef cone in the projective toric case. The cone

$E_\pi$33

is larger and tests effectivity on all orbit closures. The upper cone $E_\pi$34 is defined by codimension-one orbit tests, and for pure $E_\pi$35 one has

$E_\pi$36

If $E_\pi$37 is projective toric, then all three coincide and equal $E_\pi$38 (Gibney et al., 2010).

The main theorem states that for any projective $E_\pi$39,

$E_\pi$40

If in addition $E_\pi$41 satisfies

$E_\pi$42

and

$E_\pi$43

is surjective, then

$E_\pi$44

where $E_\pi$45 is induced by the tropical multiplicities. Hence

$E_\pi$46

This is a genuine lower-upper sandwich for the nef cone (Gibney et al., 2010).

The framework generalizes two established exact descriptions. For Mori dream spaces,

$E_\pi$47

so the lower bounds are exact. For $E_\pi$48, the upper cone $E_\pi$49 is exactly the cone of $E_\pi$50-divisors, making the inclusion

$E_\pi$51

the “only if” direction of the $E_\pi$52-conjecture (Gibney et al., 2010). In this usage, “lower-upper” describes a computational approximation package attached to a projective variety via toric and tropical geometry.

6. Upper-motive and upper-cluster perspectives

Two adjacent strands of the literature extend the lower-upper vocabulary beyond the preceding explicit usages. The first is motivic. For a suitable homology theory $E_\pi$53 and a complete variety $E_\pi$54, one defines

$E_\pi$55

For complete regular varieties, this invariant is preserved under $E_\pi$56-equivalence; equivalently, when $E_\pi$57-equivalence is detected by upper Chow motives with $E_\pi$58-coefficients, the invariant depends only on the upper motive (Haution, 2012). In particular, the classical invariants

$E_\pi$59

fit this scheme, and one obtains the lower bound

$E_\pi$60

when $E_\pi$61 (Haution, 2012). This suggests a broader interpretation in which upper structures control compressibility and splitting behavior, especially for projective homogeneous varieties.

The second strand is cluster-theoretic. For any symmetrizable Kac–Moody type, the coordinate ring of an open Richardson variety $E_\pi$62 is an upper cluster algebra: $E_\pi$63 Likewise, the coordinate ring of the generalized open Richardson variety $E_\pi$64 in a twisted product of flag varieties is an upper cluster algebra, covering reduced double Bruhat cells, Bott–Samelson varieties, and braid varieties as special cases (Bao et al., 12 Jun 2025). The lower-versus-upper cluster question is explicit here: the paper proves upper-cluster realizations, conjectures local acyclicity as a mechanism for

$E_\pi$65

and proves that equality in a product-length case (Bao et al., 12 Jun 2025).

These two directions are not themselves definitions of lower-upper varieties. They are, however, structurally adjacent. In the motivic setting, upper summands already determine numerical invariants relevant to incompressibility. In the cluster setting, varieties arise whose coordinate rings are naturally upper cluster algebras, while the lower-equals-upper problem remains a central issue. Together with the matrix, lattice, and nef-cone theories, they show that “lower-upper” has become a recurring organizing principle across several mathematically distant domains.