Hilbert series of exterior algebras
modulo two quadrics

1. Introduction

Two alternating forms define both a quotient of the exterior algebra and a family of common isotropic Grassmannians. We show that the graded pieces of the quotient are the Plücker sections of these varieties, so their geometry determines the Hilbert series and explains its combinatorial form. The calculation proves the formula conjectured by Crispin Quiñonez, Lundqvist and Nenashev [5], who expressed it by walks in a strip and established it as a coefficientwise upper bound.

Let be a complex vector space of dimension . For , put , and let count walks of length from to , with steps , staying in .

Binomial coefficients outside their ordinary range are zero, and a determinant of size zero is one. Boij and Lundqvist [4] have also determined the exterior Hilbert series for even .

The formula follows from a uniform calculation on the common isotropic Grassmannians of and . For , let

This is the zero scheme of a section of , where is the tautological bundle. For , the forms and , with , are the linear Plücker equations for common isotropy. A cohomology gap in the Koszul resolution shows that restriction of Plücker sections is surjective with precisely these relations. Hence the degree- exterior quotient is the space of Plücker sections of . The same calculation determines all higher Plücker twists. Write and . When , define the interval of -subsets

ordered componentwise.

The class of in -theory is the square of a staircase Schubert class, and hence the class of the Richardson variety for . Standard monomials on that interval then give the determinant above. Section 2 carries out this calculation.

In odd dimension the comparison is realized by a degeneration, and the exterior quotient itself has an explicit basis. For , a general pencil can be written, after changing basis and the two generators, as

The odd-dimensional geometry also determines the topology of . A regular Borel action gives the Betti numbers, and an explicit integration pairing presents and as complete intersections in tautological Chern classes, with Hilbert series . A dense vector-bundle chart proves rationality (Section 4). Returning to the exterior algebra, the monomial basis refines its Hilbert function by the two halves of the pencil, and the perfect exterior pairing recovers its two-generator Koszul homology.

The colored-edge estimate behind the cohomology gap continues to apply to several alternating forms and gives an exact range of comparison with Plücker sections. The explicit straightening rules, on the other hand, expose the characteristic dependence; their denominators give a range in which the same bases survive, while Frobenius supplies new Koszul cycles and additional quotient classes in larger dimensions.

2. The Koszul complex and a staircase class

The Koszul complex relates the exterior quotient to Plücker sections. A cohomology gap makes that comparison exact; a staircase identity in -theory then computes the section dimensions. The case is immediate, so assume . Write , , and let have dimension two. Set

The evaluation of pairs of alternating forms onto is surjective. Its incidence variety is therefore a smooth vector bundle over , of dimension

Properness and generic smoothness show that a general zero scheme is smooth of the expected dimension or empty, and is empty if .

For a regular or nowhere-zero section, the Koszul complex resolves or zero. After twisting by , its terms are in position . In its cohomology spectral sequence, an entry has total degree . Lemma 2.1 puts every entry with in total degree at most . An incoming differential to would start in total degree , so only the ordinary Koszul map can contribute. There are also no entries of positive total degree. Consequently,

Here , and the Koszul map is wedge multiplication by the two forms. The same argument gives for every ; for it also makes restriction of Plücker sections surjective.

2.1. The staircase factorization. The vanishing just proved lets us recover section dimensions from Euler characteristics. We therefore seek the -class of . The Euler class of a single alternating form turns out to be the class of a staircase Schubert variety. Let . If are the formal line-bundle roots of and , the Grassmannian Grothendieck formula [10, Theorem 7.3] is

For , set and factor from row . The remaining determinant is Vandermonde. The identity

cancels the denominator and row factors, giving

The Schubert varieties for in opposite flags meet in the Richardson variety . They have no higher Tor; apply the homological Kleiman–Bertini theorem [9] to a general translate; opposite flags form one orbit of the Borel group preserving the first Schubert variety. Consequently

The last identity is the Koszul resolution. It also proves nonemptiness for a general section in the stated range; the Richardson class has Euler characteristic one, whereas an empty Koszul zero scheme would have class zero.

2.2. Intervals, multichains and determinants. We recall an elementary form of the interval standard-monomial theorem [8].

Richardson varieties have no higher cohomology in nonnegative Plücker twists [8, Theorem 5.0.6]. The class identity and the preceding vanishing therefore identify the Hilbert function of with the multichain count. Put and

The interval is the lattice of ideals of the cell poset of . Its length- multichains count plane partitions of this shape with entries in . Kreweras’s determinant, in the form of [6, Proposition 2.2], is

Transposing and complementing the lower binomial entries gives (1).

For , the ideal of is the image of . Since

the Cartan-product map from is onto; its irreducible target has highest weight , and the product of highest-weight sections is nonzero. Thus the ideal is generated in degree one, as claimed. The complete section ring of a normal variety is integrally closed, which proves the projective-normality assertion; it applies in particular to the general smooth zero scheme.

Taking leading coefficients of the multichain determinant gives

where a negative factorial contributes zero. This is the normalized volume of the order polytope of . For , the cell poset is a fence and the degree is the Euler secant number ; the Hessenberg determinant has generating function after division by .

2.3. Duality and the negative twists. Adjunction gives . To prove Cohen–Macaulayness, it remains to treat . Write , with . For a nonzero Schur cohomology term, let be the number of negative Bott entries and . The weight argument gives , also when . To obtain the opposite bound, use . Exterior duality replaces degree by , and Serre duality on the Grassmannian replaces twist by . The complementary constituent is

with crossed Bott entries. The total cohomological degree changes from to . Applying the same weight bound to this dual term gives

The two bounds differ by . Thus every Koszul term is acyclic in this window. Serre duality and the nonnegative-twist result now give vanishing of all intermediate cohomology in every twist. The coordinate ring is the complete section ring, so it is Cohen–Macaulay, with canonical module its shift by . This proves the Gorenstein assertion of Theorem 1.2. If , the Hilbert function is identically one, so is a reduced point and the assertion is immediate.

2.4. The exterior formula. The determinant in (1) is unchanged by transposition and equals the determinant in Theorem 1.1. To identify it with walks, put

Expanding the upper Hessenberg determinant along its first row gives . For the adjacency matrix of the path on , the tridiagonal recurrence and corner cofactor give

Thus counts walks of length .

For and , a general zero scheme is empty, so (4) gives vanishing. In degree , multiplication of a nonzero coefficient term of by its complementary monomial gives the top exterior line. The case is immediate. Intersecting the finitely many open conditions proves Theorem 1.1 in all degrees simultaneously.

3. The odd pencil and its degenerations

For the odd pencil, the Hilbert-function comparison becomes a straightening procedure. A local inverse removes every monomial outside the interval ; primality of the interval ring then shows that no further relations occur. Assume . The canonical-form theorem [5, Theorem 3] puts a general pencil in the form (2). Call good if

Failure of a lower or upper inequality is called left or right failure.

3.1. An inverse on a prefix. To straighten a left-failing monomial, we first divide its initial segment by the restriction of to that segment. Set ; these elements commute and square to zero, so the required division becomes an inclusion-map calculation on subsets.

For instance, the first nontrivial inverse is

Prescribe costs on the path edges by

Choose integers with , and put . We take largest -weight as initial weight.

3.2. Primality forces equality. Let and

Give weight . Plücker relations are homogeneous for the original vertex multigrading, hence for this weight. Lemma 3.2 gives

By Lemma 2.2, the latter quotient is a domain of dimension , so this containment gives an upper bound of for . For the reverse bound, on a Grassmannian chart is defined by the pairings of its basis vectors with . It is nonempty because the odd-indexed span is common isotropic. The height theorem therefore gives . Initial ideals preserve dimension, so both bounds are equalities. A proper further quotient of the finite-type domain would lower dimension. There can be no additional initial relation, and hence

This is the required Richardson degeneration. Its degree-one basis gives the good exterior monomials; its full standard basis lifts to the actual products of good Plücker coordinates on . The rules (5) and their reflections strictly decrease weight and give unique exterior normal forms.

The path bijection is particularly direct; put down-steps at the indices of and add a fixed up-step at each end. The good inequalities are exactly the two walls of the strip. At there is one good monomial, ; above that degree every monomial has a left failure and the triangular relations span the whole exterior degree.

3.3. Toric and quadratic degenerations. Embed the Richardson ring into the band-matrix ring of Lemma 2.2. A diagonal term order gives the good minor initial monomial

Distinct standard products have distinct leading monomials, so the good minors form a SAGBI basis. The initial algebra has presentation

Indeed, these relations sort every product into a chain, whose diagonal monomial determines it uniquely.

For termination, put and give weight . Replacing an incomparable pair by its meet and join increases the separation of the two ranks without changing their sum, and strictly lowers this weight. Thus the displayed quadratics are a Gröbner basis with initial ideal .

The SAGBI step is realized by weights on the finitely many quadratic straightening relations; their greatest diagonal terms give precisely the meet-and-join binomials. Equality of standard-monomial Hilbert functions gives equality of the initial ideals. Combining these weights with (6) at separated scales gives an initial ideal generated by forbidden variables and these squarefree quadratics. Eliminating the forbidden variables proves Theorem 1.3; the quadratic Gröbner criterion gives Koszulness [11, Theorem 5.3].

4. Cohomology and rationality in odd dimension

The canonical odd pencil has a regular Borel action whose tangent weights determine the Betti numbers. Its integration pairing then recovers the ring structure, while projection onto one half of the pencil supplies a rational chart.

Throughout this section, . Write and , with bases and , so that

The canonical pair represents the general orbit, so is smooth. Put

Let and let be formal Chern roots. In half the cohomological grading, and . In the integration pairing, the Grassmannian factor combines with the Euler factor of the two isotropy conditions to give . Its annihilators come in two families; complete homogeneous polynomials in these squares, and their first variations under . Define

These are symmetric in the and thus polynomials in the .

4.1. A regular Borel action. For , define