PBW degenerate Schubert varieties: Cartan components and counterexamples
Igor Makhlin

TL;DR
This paper investigates PBW degenerations of Schubert varieties, revealing limitations in their properties and providing counterexamples, especially in the case of rak{sl}_6, through analysis of Cartan components.
Contribution
It demonstrates that certain properties of PBW degenerations do not hold universally and introduces counterexamples based on Cartan component analysis.
Findings
Counterexamples in rak{sl}_6 show limitations of PBW degenerations.
Properties depend on highest weight, not just Weyl group stabilizer.
Counterexamples challenge previous assumptions about PBW degenerations.
Abstract
In recent years PBW degenerations of Demazure modules and Schubert varieties were defined and studied in several papers. Various interesting properties (such as these PBW degenerations embedding naturally into the corresponding degenerate representations and flag varieties) were obtained in type but only with restrictions on the Weyl group element or the highest weight. We show that these properties cannot hold in full generality due to the following issue with the definition. The degenerate variety depends on the highest weight used to define it and not only on its Weyl group stabilizer (as is the case for PBW degenerate flag varieties as well as classical Schubert varieties). Perhaps surprisingly, the minimal counterexamples appear only for . The counterexamples are constructed with the help of a study of the Cartan components appearing in this context.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
PBW degenerate Schubert varieties: Cartan components and counterexamples
I. Makhlin
Skolkovo Institute of Science and Technology
Center for Advanced Studies
Bolshoy Boulevard 30, bld. 1
Moscow 121205
Russia
National Research University Higher School of Economics
International Laboratory of Representation Theory and Mathematical Physics
Ulitsa Usacheva 6
Moscow 119048
Russia
Abstract.
In recent years PBW degenerations of Demazure modules and Schubert varieties were defined and studied in several papers. Various interesting properties (such as these PBW degenerations embedding naturally into the corresponding degenerate representations and flag varieties) were obtained in type but only with restrictions on the Weyl group element or the highest weight. We show that these properties cannot hold in full generality due to the following issue with the definition. The degenerate variety depends on the highest weight used to define it and not only on its Weyl group stabilizer (as is the case for PBW degenerate flag varieties as well as classical Schubert varieties). Perhaps surprisingly, the minimal counterexamples appear only for . The counterexamples are constructed with the help of a study of the Cartan components appearing in this context.
The author would like to thank Lara Bossinger, Xin Fang, Evgeny Feigin, Ghislain Fourier and Ievgen Makedonskyi for helpful discussions of these subjects. The work was partially supported by the grant RSF 19-11-00056. This research was supported in part by Young Russian Mathematics award.
Introduction
Over the last decade PBW degenerations of representations and flag varieties proved to be a diverse and fruitful research topic ([Fe, FFL1, ABS, FFL2, CFR, CL] and many others). Let us briefly outline the definitions of these objects.
Consider a semisimple Lie algebra and the subalgebra spanned by negative root vectors, an integral dominant -weight and the irreducible representation . The PBW filtration on defines a filtration on via action on the highest weight vector, the associated graded space is the PBW degenerate representation (of the abelian Lie algebra ). The space is then seen to be acted upon by the abelian Lie group , the closure of the -orbit of the highest weight point in is the corresponding PBW degenerate flag variety.
Let us restrict our attention to algebras of type . Here, a fundamental property of the degenerate flag varieties which makes them especially interesting is that they depend on the highest weight the same way as flag varieties do themselves. Let for fundamental weights , then is determined by the Weyl group stabilizer of , i.e. the set of such that . This was first proved in [Fe] by constructing a degenerate Plücker embedding of sorts for . A highly important fact, from which the Plücker embedding is obtained, is that can be realized as the cyclic submodule generated by the highest weight vector in
[TABLE]
i.e. the so-called Cartan component.
Now, a very natural generalization here seems to be the transition from irreducible representations and flag varieties to Demazure modules and Schubert varieties. For a Weyl group element the Demazure module is generated by a lowest weight vector. This allows us to define its PBW degeneration and the orbit closure in analogy with the above.
These objects were, apparently, first considered in [Fo] where many of the key properties of and were generalized to the case of a triangular , i.e. a permutation avoiding the patterns and . It was shown that for such the variety admits a degenerate Plücker embedding and, consequently, depends on the same way the Schubert (or flag) variety does. The mentioned Cartan component property was also generalized to this case and a generalization of the FFLV basis ([FFL1]) was constructed. Let us point out that, in a sense, the underlying cause for this neat picture is the fact that the PBW filtration on can be induced from the one on . This realizes as the a cyclic submodule in over a certain subalgebra in and also realizes as an orbit closure in for certain subgroup in (see also Remark 2.8).
In the subsequent paper [BF] similar properties are obtained in the case of a minuscule highest weight and arbitrary . The most recent paper [CFF] on this subject proved that when is rectangular (a subclass of the triangular elements), is itself a Demazure module and is itself a Schubert variety for some algebra of higher rank. This generalizes analogous properties of and proved in [CL].
Little, however, has been proved concerning type PBW degenerate Demazure modules and Schubert varieties of full generality. Nevertheless, some of the experts in this field expected the degenerate Plücker embedding or, at least, the mentioned independence of to generalize to the case of an arbitrary (as was communicated to the author during some private and public discussions). We show that this is not the case proving that neither of these results hold in full generality. Perhaps the most surprising aspect here is that the minimal counterexamples appear only for which makes spotting them a challenge even with the use of a computer. As discussed in Remark 2.16, for the Cartan component property has been verified for all which implies both the Plücker embedding and the independence of . This circumstance lead the author to put a considerable amount of effort into proving the same for greater only to discover said counterexamples after gradually optimizing his search algorithms.
Prior to presenting the counterexamples in Section 3 we provide a study of the Cartan component and the orbit closure subvariety in . It turns out that these objects are somewhat easier to control than and but can be tied to the latter via a homomorphism between and . These results are then applied to the construction of counterexamples. It should be pointed out that the proofs in Section 3 require some computer assistance (the Demazure modules in question have dimensions 2942 and 8226). This, however, is kept to a bare minimum, i.e. finding Minkowski sums of certain small sets: a computation which is easily reproducible.
1. Preliminaries
We first recall the definitions and key properties of the more conventional abelian PBW degenerations of -representations and flag varieties and then define the analogous objects for Demazure modules and Schubert varieties.
1.1. PBW degenerate representations and flag varieties
For consider the Lie algebra with a fixed triangular decomposition . For denote the simple roots and let be the corresponding fundamental weights. The set of positive roots is comprised of
[TABLE]
for (the set of negative roots is denoted ). Let be spanned by negative root vectors with weight and be spanned by positive root vectors with weight . Our basis of choice in will be the set of fundamental weights, i.e. will denote the weight . The semigroup of integral dominant weights (those with coordinates in ) will be denoted .
The universal enveloping algebra is equipped with a -filtration (the PBW filtration) with components
[TABLE]
The associated -graded algebra is a polynomial algebra in variables (the images of in ). The algebra can be viewed as for an abelian Lie algebra spanned by the .
For let be the irreducible representation of with highest weight and highest weight vector (so that ). The PBW filtration induces a -filtration on every such by . The associated graded space is naturally a -module known as the (abelian) PBW degeneration of . This module is cyclic, generated by a vector (the image of in ).
Now consider the Lie group with and the nilpotent subgroup with . Choose with the tuple of integers for which . The tuple determines the parabolic subgroup preserving the line . This line provides a point in the projectivization . The stabilizer is seen to be , therefore the orbit closure is the partial flag variety (with the orbit itself being the largest Bruhat cell).
Similarly, we may consider the connected simply connected Lie group with (this Lie group will simply be under addition). The group acts on and . We have corresponding to and we define — the (abelian) PBW degeneration of . The fact that depends only on and not on itself is a consequence of Theorem 1.1 (the degenerate Plücker embedding) which we now state.
Let the -dimensional complex space be the standard representation of with basis . The irreducible representations with fundamental highest weights can be explicitly described as with a basis consisting of the vectors
[TABLE]
We may assume that .
Consider the Plücker embedding
[TABLE]
The product is equipped with the Plücker coordinates with , coordinate corresponding to . The homogeneous coordinate ring of is . The homogeneous coordinate ring of is then , where is the ideal of Plücker relations.
Now we introduce a grading on by setting
[TABLE]
Let us note that this grading has a straightforward interpretation: is the least such that . For instance, we have which is the PBW degree of the monomial mapping to .
We proceed to consider the corresponding initial ideal , i.e. the ideal spanned by components of minimal grading of elements of .
Theorem 1.1** ([Fe, Theorem 3.13]).**
is isomorphic to the subvariety in cut out by the ideal .
We will also require another (closely related) fact from this theory.
Definiton 1.2**.**
Given a finite set of cyclic modules over a Hopf algebra with respective generators , the submodule in generated by is known as the Cartan component.
Let us consider two instances of this very general notion. For the chosen consider the tensor product
[TABLE]
It is well known that the Cartan component in generated by
[TABLE]
is isomorphic to the irreducible representation (simply because has weight and is the unique highest weight vector in ). A similar fact holds for PBW degenerations. Denote
[TABLE]
and
[TABLE]
Theorem 1.3** (essentially due to [FFL1] but see also [FFL3, Theorem 10.4]).**
The Cartan component is isomorphic to .
Next, let us explain the nature of the embedding . On one hand, in view of Theorem 1.3, is isomorphic to where corresponds to the line . On the other, we have the embeddings
[TABLE]
where the first embedding is diagonal and the second is the Segre embedding. The image of under this embedding contains and this image is preserved by the -action on , therefore the orbit and its closure are contained in .
This lets us characterize the ideal as the kernel of a certain map from to a polynomial ring. Choose for and consider
[TABLE]
Definiton 1.4**.**
For and a tuple consider the vector and decompose it into a linear combination of the . Denote the polynomial (homogeneous of degree ) such that the coefficient of is equal to .
One sees that such a polynomial exists by expanding into a (terminating) series.
Proposition 1.5** (see [FaFFM, Proof of Theorem 3.4]).**
The ideal is the kernel of the homomorphism from to taking to .
1.2. PBW degenerate Demazure modules and Schubert varieties
The Weyl group acts on , the permutation maps to where we set if . For every there is a unique (up to a scalar multiple) vector of weight .
Definiton 1.6**.**
The -module is the corresponding Demazure module.
Definiton 1.7**.**
Let be the subalgebra spanned by with (i.e. ).
Note that for every positive root such that we have . Consequently, . The dimension of is the permutation length .
Let correspond to and while . The Schubert variety is the orbit closure . This orbit closure indeed depends only on and and is naturally embedded into since is contained in .
Now, like above, we have the PBW filtration on with component spanned by PBW monomials of degree no greater than . This induces the filtration .
Definiton 1.8**.**
The associated graded algebra for the former filtration is for an abelian Lie algebra . The associated graded space for the latter filtration is , the PBW degeneration of the Demazure module.
The algebra is spanned by which are images of . The space is a module over . Furthermore, is acted upon by the corresponding Lie group (i.e. under addition) and so is . Let be generated by , let the point correspond to .
Definiton 1.9**.**
is the PBW degeneration of the Schubert variety .
It should be noted that we could consider the PBW degeneration of the whole , view as a -module and define as a -orbit closure for the larger group . However, all act trivially on as well as the corresponding subgroups . This is why we lose nothing by limiting our attention to .
An observation due to [Fo, Subsection 1.2] is that it is useful to consider a shift of this construction by . Recall that the group acts on by automorphisms that preserve while permuting the root subspaces. This induces a -action on every .
Definiton 1.10**.**
Consider the subalgebra .
By the definition of , the subalgebra is contained in and is spanned by such that . In fact, we have while . The space
[TABLE]
will also be a Demazure module but with respect to a different Borel subalgebra: not but .
Once again, we have the PBW filtrations on and .
Definiton 1.11**.**
Let and (PBW degeneration of ) be the respective associated graded objects.
We can naturally identify with the subalgebra in spanned by with . The space is acted upon by and the corresponding Lie group . Let correspond to the line .
Proposition 1.12**.**
The orbit closure is isomorphic to .
Proof.
Consider the isomorphism mapping to . We can now view as a -module by letting act as . It is evident that and are isomorphic -modules. Furthermore, we have the isomorphism which defines an -action on . We see that and are isomorphic -representations and the proposition follows. ∎
To summarize the Proposition and its proof, the objects , and are isomorphic to the objects , and and can be studied instead of these latter objects. The usefulness comes from the fact that (resp. ) is naturally embedded into (resp. ) which lets us exploit our understanding of PBW degenerations of irreducible representations and flag varieties. This idea is applied in [Fo, Section 5] to show, among other things, that if is sufficiently nice (“triangular”), then depends only on and embeds naturally into . This approach is then furthered in [CFF, Theorem 2.4] to show that for a certain narrower class of permutations (termed “rectangular”) the variety is actually a Schubert variety for a certain .
2. General observations
In this section we will derive some general facts about the PBW degenerations and via the study of the corresponding Cartan components. These facts will be useful to us in the construction of our counterexamples.
Within this section fix and with . Since and are defined as associated graded spaces, they are equipped with a -grading that we also denote . For a -graded space we will simply write to denote its -homogeneous component of grading . Also observe that, since is embedded into , we have a -module structure on .
Proposition 2.1**.**
There exists a homomorphism of -modules
[TABLE]
that respects and takes to .
Proof.
The Demazure module is defined as a subspace of which provides an embedding (PBW filtrations with respect to and respectively) for all . These embeddings induce maps which sum up to give . ∎
Definiton 2.2**.**
Denote the image of .
In other words, .
Definiton 2.3**.**
Denote the variety .
Since is embedded in to , the variety is embedded into and depends only on , hence the notation. The modules and are easily described.
Proposition 2.4**.**
The maps are injective for all and . The subspace is spanned by the vectors for which the following holds. For any the th smallest number among is no greater than the th smallest number among .
Proof.
The vectors with the property from the last sentence of the proposition are known to span . Therefore, for dimensional reasons, one is only to verify that each such is indeed contained in . Such a is obtained from by the action of the product of all such that is the th smallest number among and is the th smallest number among for some and, moreover, . ∎
Remark 2.5**.**
The module is embedded into . Therefore is the -submodule in the tensor product
[TABLE]
generated by the highest weight vector. This lets one view as the Cartan component for PBW degenerations of Demazure modules.
Definiton 2.6**.**
Let be the rational map induced by .
Proposition 2.7**.**
The map restricts to a birational equivalence between and .
Proof.
For the vector is the image of in , hence if and only if . Similarly, if and only if , therefore and have the same stabilizer in . Since is a homomorphism of -modules, the map identifies the orbits and and thus provides a birational equivalence between their closures. ∎
Remark 2.8**.**
It is shown in in [Fo, Subsection 3.4] that if is triangular, then the map is injective. The map , therefore, is an injective morphism and provides an isomorphism between and . Part of our goal in this paper is showing that this does not necessarily happen for general and we may, moreover, have .
The defining ideal of can be characterized algebraically.
Definiton 2.9**.**
Let be the ideal cutting out .
Definiton 2.10**.**
Denote the polynomial obtained from the polynomial defined in Subsection 1.1 by setting when .
Proposition 2.11**.**
The ideal is the kernel of the homomorphism from to taking to .
Proof.
Choose numbers for all pairs with (i.e. ) and consider . Choose some and a tuple , consider the vector and decompose it into a linear combination of the . The coefficient of will equal . This (together with the bijectivity of the exponential map for ) implies the proposition. ∎
Before proceeding we observe that the ring is graded by by setting . The defining ideals of subvarieties in are -homogeneous and so are their initial ideals. For a -graded space we will write to denote its -homogeneous component of grading .
Now let us give a slight modification of Theorem 1.3. Denote
[TABLE]
and
[TABLE]
Since is embedded into as the space of tensors that are symmetric in the according sense, we immediately see that the -submodule is isomorphic to .
The space has a basis comprised of vectors obtained by taking for each a (symmetric) product of vectors of the form and then taking the tensor product of these symmetric products. The component has a basis obtained from this basis in by replacing each with and symmetric and tensor products by products in the ring . The bijection between these two bases establishes a duality between and .
Proposition 2.12**.**
The subspace is the orthogonal of the subspace with respect to the above duality.
Proof.
We have identified with the space of linear forms on , i.e. . Note that the image of the embedding is contained in . By the definition of the Segre embedding, vanishes as a linear form in a point in , if and only if vanishes as polynomial in the Plücker coordinates of .
Let be the subspace in dual to . Then is the minimal projective subspace in which contains but this is precisely because is cyclic. ∎
Next, the subgroup defines a Schubert variety , this subvariety is isomorphic to but is embedded differently into and, subsequently, . The embedding provides an ideal .
Also note that the PBW filtration on induces a filtration on , the associated graded space is a -submodule in that contains .
Proposition 2.13**.**
The subspace is the orthogonal of the subspace .
Proof.
We may consider the space
[TABLE]
with the linear isomorphism which identifies the corresponding products of the and the . We obtain a -grading on via and a filtration . We see that maps to , therefore the associated graded space is a -module. By comparing the action of on and , we see that the two are isomorphic as -modules. The fact that the highest weight vector in generates while that in generates implies that .
From the above we see that the subspace is spanned by all vectors of the form where and is the nonzero -homogeneous component of with the largest grading. Consequently, is spanned by with .
The above description of together with the fact that is dual to implies that and do indeed annihilate each other. The proposition then follows for dimensional reasons. ∎
We can now embed into a Gröbner degeneration of (i.e. a variety cut out by an initial ideal of ).
Proposition 2.14**.**
We have , i.e. is a subvariety of the variety cut out by . The map is injective if and only if .
Proof.
The first claim is immediate from Propositions 2.12 and 2.13 in view of . Now, the map is injective if and only if . However, and Propositions 2.12 and 2.13 imply
[TABLE]
The second claim follows. ∎
Now we can prove the following.
Corollary 2.15**.**
The following are equivalent.
- (a)
The map is injective for all with . 2. (b)
. 3. (c)
is prime.
Proof.
By the second part of Proposition 2.14, (a) is equivalent to (b). Since is irreducible (as an orbit closure), (b) implies (c).
Now, suppose that is prime. On one hand, the stabilizers and have the same dimension (see proof of Proposition 2.7), and so do the orbit closures and . On the other, denote the variety cut out by . By a standard construction (see, for instance, [HH, Corollary 3.2.6]), we have a flat family over with the fiber over 0 being and with all other fibers being . By flatness we have but we know that which together with the irreducibility of both implies (b). ∎
Remark 2.16**.**
Condition (c) in the above theorem has been verified for and all with the use of the computer algebra system Macaulay2 ([M2]). However, as will be shown below, the conditions (a), (b) and (c) may fail when . It should be pointed out that computing initial ideals and checking them for primeness appears to be rather resource intensive when . Instead, the counterexamples in Section 3 were found with the use of a more complicated program written in SageMath ([S]). This program constructed a basis in incrementally by emulating the actions of the on the space and then compared the dimensions of and .
Before we proceed with our counterexamples let us discuss a construction that generalizes FFLV bases and the monomial bases obtained in [Fo, Theorem 1]. The below strategy of constructing monomial bases (or, at least, linearly independent subsets) is standard in the theory of PBW degenerations.
Denote the bijection from the set of monomials in the (spanning ) to the semigroup which takes a monomial to its exponent vector. Choose a total monomial order on the universal enveloping algebra, this is the same as choosing a total semigroup order (also ) on the semigroup.
Now, is spanned by vectors of the form . For every choose the -minimal monomial such that is a nonzero multiple of . Let be the set of for all (this set has size ).
Define the Minkowski sum
[TABLE]
And consider the set of monomials .
Proposition 2.17**.**
The set is linearly independent.
Proof.
By Theorem 1.3 we are to show that the set is linearly independent. Consider the subspace
[TABLE]
evidently . We may define a -grading on by letting the grading of any tensor product of the be the sum of the corresponding . Now, if we take and expand into a sum of such tensor products, the summands in the result will either lie in the -homogeneous component of grading or in a component of some other grading with . At least one of the former summands occurs with a nonzero coefficient and the linear independence follows. ∎
We fix a specific order which from now on will be denoted , the definitions of and will now be understood with respect to this order. This order will be a graded lexicographic order with respect to a certain ordering of all . This means that for two monomials we have if and only if or and the first in our ordering that and contain in unequal degrees is contained in in the lesser degree. In the ordering we choose the are sorted by increasing and within a fixed by increasing.
Remark 2.18**.**
For the set provides the FFLV basis in ([FFL1, Theorem 1.5]) and, more generally, for triangular this gives the basis constructed in [Fo, Theorem 1]. In fact, for and all with coordinates no greater than 3 it has been verified computationally that and, therefore, the monomials in provide a basis in . One could speculate that this always holds when but, again, we will show that this fails when . Furthermore, in the cases we consider we will have . This, however, fails for and .
3. Counterexamples
Within this section we set
[TABLE]
where is written in one-line notation, i.e. as .
We say that two varieties with a -action are isomorphic as -varieties if their exists an -equivariant isomorphism between them. Our main theorem, to the proof of which we devote this section is as follows.
Theorem 3.1**.**
is not isomorphic to as a -variety.
Remark 3.2**.**
Throughout the theory of PBW degenerations the degenerate varieties are defined as orbit closures and considered together with the action of the corresponding group, therefore we consider Theorem 3.1 to be sufficient for our purposes. However, SageMath computations have shown that the number of points fixed by the action of a certain torus is different in and in (see also Remark 3.7). This shows that and have different Euler characteristics and are not isomorphic as algebraic varieties and even as topological spaces.
Our strategy of showing that two varieties are not isomorphic will, nonetheless, be to compare the actions of the mentioned torus, let us introduce these actions. The spaces , , and admit natural weight space decompositions, i.e. are equipped with an -action and an action of the maximal torus . Furthermore, the grading induces an additional -action by acting as . We obtain an action of the -dimensional torus .
Now, the Lie algebra is also acted upon by (where all ) and so is the Lie group . This means that the spaces , , and and their projectivizations are acted upon by the semidirect product . Since fixes , and , the open -orbits in , and are -invariant and so are the orbit closures themselves.
Proposition 3.3**.**
If and are isomorphic as -varieties, then there exists a -equivariant isomorphism between them that is also -equivariant and maps to .
Proof.
Suppose that is an -equivariant isomorphism. Since each of the varieties contains a single open -orbit, we have for some . Now consider the map with . Since is abelian, will be a -equivariant isomorphism mapping to .
Let us show that is also -equivariant. Indeed, the action of on is given by for and (and ). The same formula holds for replacing and, since for any , we see that is -equivariant when restricted to the open orbit. The claim follows. ∎
For a tuple of complex numbers denote . For a choose a basis in consisting of -weight vectors and consider the coordinates of . The coordinate corresponding to a vector in our basis will be a homogeneous polynomial in the of degree . For a generic none of these coordinates vanish.
Consider the maximal such that the -homogeneous component is nonzero. One may define the limit of as approaches infinity. The homogeneous coordinate of this point corresponding to a vector in the chosen bases will be zero if it will equal the corresponding polynomial in the (or any ) whenever . We denote this point . For generic in the above sense, the point is -fixed if and only if all vectors in have the same -weight. We now fix a that is generic in this sense for both and . We will show that is -fixed while is not.
To show that is -fixed we show that while all with are zero. This will be done by identifying this one-dimensional subspace with the kernel of . The first step will be showing that , i.e. that . This is where we will require a certain amount of computer assistance.
First of all, we note that . Tools for finding dimensions of Demazure modules can be found in LiE ([LiE]) and SageMath ([S]) and provide .
Lemma 3.4**.**
The dimension of is at least 2941.
Proof.
The size of the set as defined in Section 2 can be computed to equal 2941 and the lemma follows from Proposition 2.17.
For the sake of completeness we write out the relevant sets with the use of Proposition 2.4.
[TABLE]
Herefrom the size of the set is easily found to be 2941 with a computing system or programming language of choice. ∎
Lemma 3.5**.**
We have . This kernel coincides with while all with are zero.
Proof.
Our first step will be showing that which together with Lemma 3.4 will imply the first claim. We do this by making use of Proposition 2.14, i.e. by producing an element in . Explicitly, this element is
[TABLE]
The fact that is verified using Proposition 2.11. One easily finds that , , , , , , , .
To show that we make use of another “weight” grading on . This -grading is given by the th coordinate of being equal to 1 if and 0 otherwise. Note that such a grading can be viewed as a -action on . The image of the diagonal embedding then acts trivially and we obtain an action of which is the action of the maximal torus . Hence, the ideals we consider are -homogeneous.
Now, the ideal is known (see, for instance, [KM, Remark 9]) to be generated by and the variables for which there is an such that the th smallest number among is greater than the th smallest number among . (In other words, such that ). In our case we obtain two such variables: and .
However, it is easily checked that there are no monomials in that have the same -grading as and are divisible by either or . This means that if we have , then we must have . But the latter is disproved with the use of Proposition 1.5. We have which divides none of , , and .
We have shown that the kernel is one-dimensional and we let be a spanning vector.
Now, since we see that . In fact, Propositions 2.12 and 2.13 imply that
[TABLE]
Hence
[TABLE]
while for all other . Consequently,
if and only if .
Denote . By the definition of , the fact that means that the component of the PBW filtration with respect to is strictly smaller than (component of the PBW filtration with respect to ). The fact that is injective when restricted to any with means that . Setting we obtain
[TABLE]
In view of the above this immediately provides .
Finally, observe that the and hence whenever . This is since the highest grading occurring in is additive with respect to (due, for instance, to the Minkowski sum property of the FFLV polytopes, see [FFL1, Proposition 3.7]) and for this highest grading is (since that is the maximal value of the right-hand side of (1)). Therefore, for any with we have , i.e. and its multiples are the only such . ∎
Now let us see what can be said about .
Lemma 3.6**.**
We have . This kernel coincides with and is spanned by 5 vectors with pairwise distinct -weights.
Proof.
First, in the spirit of the proof of Lemma 3.4 the size of the set is computed to equal 8221. One may further compute that . We see that and .
Replacing the vector with a scalar multiple of itself if necessary, we may choose a monomial of degree 7 such that . To prove the lemma it now suffices to show that the vectors , , , and are nonzero and lie in .
Choose a . First we show that is nonzero. Choose a monomial in obtained from by removing the a superscipts and consider the product of -modules . The cyclic submodule therein generated by is isomorphic to . We are to show that may not be expressed as a linear combination of vectors of the form with . Let us consider a basis in comprised of and a basis in . Together with the basis consisting of in this provides a basis in the tensor product. The decomposition of with respect to our basis is then seen to contain with a nonzero coefficient. However, the decomposition of every with contains with a zero coefficient.
The fact that simply follows from . ∎
Theorem 3.1 is now immediate.
Proof of Theorem 3.1.
The discussion preceding Lemma 3.4 together with Lemmas 3.5 and 3.6 shows that is -fixed while is not. ∎
Remark 3.7**.**
Although not immediate from the above, it is true that neither of and coincides with . In fact, the rational map establishes a bijection between the -fixed points in and the -fixed points in . The same holds for the map and the -fixed points in . This means that if the number of -fixed points in is , then the number of -fixed points in is and the number of -fixed points in is . However, proving this requires substantially more computer assistance, for this reason the above argument was chosen.
Remark 3.8**.**
We have seen that the existing definition of abelian PBW degenerations of Schubert varieties has a significant disadvantage: these varieties do not depend on the highest weight the way one would expect, i.e. the way Schubert varieties do themselves. However, this dependence may still be worth an investigation. In particular, one fairly natural and interesting question arises: might it be that if we fix while letting approach infinity in some sense, then will stabilize letting us define the PBW degeneration as this limit form?
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[ABS] F. Ardila, T. Bliem, D. Salazar, Gelfand–Tsetlin polytopes and Feigin-Fourier-Littelmann-Vinberg polytopes as marked poset polytopes , Journal of Combinatorial Theory, Series A 118 (2011), no. 8, 2454–2462.
- 2[BF] R. Biswal, G. Fourier, Minuscule Schubert Varieties: Poset Polytopes, PBW-Degenerated Demazure Modules, and Kogan Faces , Algebras and Representation Theory, 18 (2015), 1481–1503.
- 3[CFF] R. Chirivi’, X. Fang, G. Fourier, Degenerate Schubert Varieties in Type A , ar Xiv:1808.01594.
- 4[CFR] G. Cerulli Irelli, E. Feigin, M. Reineke, Quiver Grassmannians and degenerate flag varieties , Algebra & Number Theory 6 (2012), no. 1, 165–194.
- 5[CL] G. Cerulli Irelli, M. Lanini, Degenerate Flag Varieties of Type A and C are Schubert Varieties , International Mathematics Research Notices 2015 , no. 15, 6353–6374.
- 6[Fa FFM] X. Fang, E. Feigin, G. Fourier, I. Makhlin, Weighted PBW degenerations and tropical flag varieties , Communications in Contemporary Mathematics 21 (2019), no. 1, 1850016.
- 7[Fe] E. Feigin, 𝔾 a M superscript subscript 𝔾 𝑎 𝑀 {\mathbb{G}}_{a}^{M} degeneration of flag varieties , Selecta Mathematica, New Series 18 (2012), no. 3, 513–537.
- 8[FFL 1] E. Feigin, G. Fourier, P. Littelmann, PBW filtration and bases for irreducible modules in type A n subscript 𝐴 𝑛 {A}_{n} , Transformation Groups 16 (2011), no. 1, 71–89.
