A SAGBI BASIS FOR THE QUANTUM GRASSMANNIAN

FRANK SOTTILE AND BERND STURMFELS

Dedicated to the memory of Gian-Carlo Rota

Abstract. The maximal minors of a p×(m+p)-matrix of univariate polynomials of degree n with indeterminate coefficients are themselves polynomials of degree np. The subalgebra generated by their coefficients is the coordinate ring of the quantum Grassmannian, a singular compactification of the space of rational curves of degree np in the Grassmannian of p-planes in (m + p)-space. These subalgebra generators are shown to form a sagbi basis. The resulting flat deformation from the quantum Grassmannian to a toric variety gives a new “Gröbner basis style” proof of the Ravi-Rosenthal-Wang formulas in quantum Schubert calculus. The coordinate ring of the quantum Grassmannian is an algebra with straightening law, which is normal, Cohen-Macaulay, Gorenstein and Koszul, and the ideal of quantum Plücker relations has a quadratic Gröbner basis. This holds more generally for skew quantum Schubert varieties. These results are well-known for the classical Schubert varieties (n = 0). We also show that the row-consecutive p × p-minors of a generic matrix form a sagbi basis and we give a quadratic Gröbner basis for their algebraic relations.

1. Statement of the main result

Let M(t) be the p × (m + p)-matrix whose i, jth entry is the degree n polynomial in t,

x_{i,j}^{(n)} · t^{n} + x_{i,j}^{(n−1)} · t^{n−1} + … + x_{i,j}^{(2)} · t^{2} + x_{i,j}^{(1)} · t + x_{i,j}^{(0)}.

The coefficients x_{i,j}^{(l)} are indeterminates. We write k[X] for the polynomial ring over a field k generated by these indeterminates, for i = 1, …, p, j = 1, …, m + p, and l = 0, …, n. Lexicographic order on the triples l, i, j gives a total order of these variables. For example,

x_{1,2}^{(0)} < x_{1,2}^{(1)} < x_{1,5}^{(1)} < x_{2,3}^{(1)} < x_{2,4}^{(1)} < x_{1,3}^{(2)} .

Let ≺ be the resulting degree reverse lexicographic term order on the polynomial ring k[X].

For each α ∈ \binom{[m+p]}{p} and a ≥ 0, let α^{(a)} be a variable which, when a ≤ np, formally represents the coefficient of t^{a} in the maximal minor of M(t) given by the columns

1991 Mathematics Subject Classification. 13P10, 13F50, 14M12, 14M15, 14M17. Key words and phrases. straightening law, poset, quantum cohomology, Schubert calculus, Grass-

mannian, Gröbner basis, sagbi basis.

Second author supported in part by NSF grant DMS-9796181. Research at MSRI supported in part by NSF grant DMS-9701755.

indexed by α_{1}, α_{2}, …, α_{p}. These variables α^{(a)} have a natural partial order, denoted C_{p,m}, which is defined as follows:

α^{(a)} ≤ β^{(b)} ⟺ a ≤ b and α_{i} ≤ β_{b−a+i} for i = 1, 2, …, p − b + a.

Fix 0 ≤ q ≤ np and let C_{p,m}^{q} denote the truncation of the infinite poset C_{p,m} to the finite subset \{ α^{(a)} | α ∈ \binom{[m+p]}{p} and a ≤ q \}. The posets C_{p,m}^{q} are graded distributive lattices. Figure 1 shows C_{2,3}^{1}.

Figure 1. The distributive lattice C_{2,3}^{1}.

Let φ : k[C_{p,m}^{q}] → k[X] denote the k-algebra homomorphism which sends the formal variable α^{(a)} to the coefficient of t^{a} in the αth maximal minor of the matrix M(t). For example,

φ(456^{(2)}) = coefficient of t^{2} in det [ \begin{matrix}x_{1,4}^{(0)} + x_{1,4}^{(1)}t & x_{1,5}^{(0)} + x_{1,5}^{(1)}t & x_{1,6}^{(0)} + x_{1,6}^{(1)}t \\ x_{2,4}^{(0)} + x_{2,4}^{(1)}t & x_{2,5}^{(0)} + x_{2,5}^{(1)}t & x_{2,6}^{(0)} + x_{2,6}^{(1)}t \\ x_{3,4}^{(0)} + x_{3,4}^{(1)}t & x_{3,5}^{(0)} + x_{3,5}^{(1)}t & x_{3,6}^{(0)} + x_{3,6}^{(1)}t \end{matrix} ]

= − \underline{x_{3,6}^{(0)}x_{1,5}^{(1)}x_{2,4}^{(1)}} + x_{3,5}^{(0)}x_{1,6}^{(1)}x_{2,4}^{(1)} + x_{3,6}^{(0)}x_{1,4}^{(1)}x_{2,5}^{(1)} − x_{3,4}^{(0)}x_{1,6}^{(1)}x_{2,5}^{(1)} − x_{3,5}^{(0)}x_{1,4}^{(1)}x_{2,6}^{(1)} + x_{3,4}^{(0)}x_{1,5}^{(1)}x_{2,6}^{(1)}

+ x_{2,6}^{(0)}x_{1,5}^{(1)}x_{3,4}^{(1)} − x_{2,5}^{(0)}x_{1,6}^{(1)}x_{3,4}^{(1)} − x_{2,6}^{(0)}x_{1,4}^{(1)}x_{3,5}^{(1)} + x_{2,4}^{(0)}x_{1,6}^{(1)}x_{3,5}^{(1)} + x_{2,5}^{(0)}x_{1,4}^{(1)}x_{3,6}^{(1)} − x_{2,4}^{(0)}x_{1,5}^{(1)}x_{3,6}^{(1)}

− x_{1,6}^{(0)}x_{2,5}^{(1)}x_{3,4}^{(1)} + x_{1,5}^{(0)}x_{2,6}^{(1)}x_{3,4}^{(1)} + x_{1,6}^{(0)}x_{2,4}^{(1)}x_{3,5}^{(1)} − x_{1,4}^{(0)}x_{2,6}^{(1)}x_{3,5}^{(1)} − x_{1,5}^{(0)}x_{2,4}^{(1)}x_{3,6}^{(1)} + x_{1,4}^{(0)}x_{2,5}^{(1)}x_{3,6}^{(1)} .

Theorem 1. The set of polynomials φ(α^{(a)}) as α^{(a)} runs over the poset C_{p,m}^{q} forms a sagbi basis with respect to the reverse lexicographic term order ≺ on k[X] defined above.

Our second theorem states that the subalgebra image(φ) of k[X] generated by this sagbi basis is an algebra with straightening law on the poset C_{p,m}^{q}. Let ≺ be the degree reverse lexicographic term order on k[C_{p,m}^{q}] induced by any linear extension of the poset C_{p,m}^{q}. This term order on k[C_{p,m}^{q}] and the previous term order on k[X] are fixed throughout this paper.

Theorem 2. The reduced Gröbner basis of the kernel of φ consists of quadratic polynomials in k[C_{p,m}^{q}] which are indexed by pairs of incomparable variables γ^{(c)}, δ^{(d)} in the poset C_{p,m}^{np},

S(γ^{(c)}, δ^{(d)}) = γ^{(c)} · δ^{(d)} − (γ^{(c)} ∨ δ^{(d)}) · (γ^{(c)} ∧ δ^{(d)}) + lower terms in ≺,

and all lower terms λβ^{(b)}α^{(a)} in S(γ^{(c)}, δ^{(d)}) satisfy β^{(b)} < γ^{(c)} ∧δ^{(d)} and γ^{(c)} ∨δ^{(d)} < α^{(a)}.

The join ∨ and meet ∧ appearing in the above formula are the lattice operations in C_{p,m}^{np}. The combinatorial structure of this distributive lattice will become clear in Section 2, when we introduce the toric variety associated with C_{p,m}^{np}. In Section 3 we interpret the subalgebra image(φ) of k[X] as the coordinate ring of the quantum Grassmannian. Section 4 contains the proofs of Theorems 1 and 2. These results generalize the classical sagbi basis property of maximal minors [19, Theorem 3.2.9] and its geometric interpretation as a toric deformation [20, Proposition 11.10] from the case of the Grassmannian to the quantum Grassmannian. In Section 5 we discuss corollaries, applications and some open problems. One such application is that the row-consecutive p × p-minors of any matrix of indeterminates form a sagbi basis.

We thank Aldo Conca, Ezra Miller, and Brian Taylor for their helpful comments.

2. The toric variety of the distributive lattice

Theorem 1 asserts that the initial algebra of our subalgebra image(φ) is generated by the initial monomials of its generators φ(α^{(a)}). Our first step is to identify the initial monomials. Here are two examples. The first one is the underlined monomial right before Theorem 1:

in_{≺}(φ(456^{(2)})) = x_{3,6}^{(0)}x_{1,5}^{(1)}x_{2,4}^{(1)} and in_{≺}(φ(2457^{(5)})) = x_{2,7}^{(1)}x_{3,5}^{(1)}x_{4,4}^{(1)}x_{1,2}^{(2)}.

In general, the initial monomial of φ(α^{(a)}) is given by the following lemma:

Lemma 3. Let α ∈ \binom{[m+p]}{p} and a = pl + r with integers p > r ≥ 0. Then

in_{≺}(φ(α^{(a)})) = x_{r+1,α_{p}}^{(l)}x_{r+2,α_{p−1}}^{(l)} ⋯ x_{p,α_{r+1}}^{(l)}x_{1,α_{r}}^{(l+1)}x_{2,α_{r−1}}^{(l+1)} ⋯ x_{r,α_{1}}^{(l+1)}.

Proof. Let x_{i_{1},j_{1}}^{(l_{1})}x_{i_{2},j_{2}}^{(l_{2})} ⋯ x_{i_{p},j_{p}}^{(l_{p})} be a monomial which appears in φ(α^{(a)}). We claim that

x_{i_{1},j_{1}}^{(l_{1})}x_{i_{2},j_{2}}^{(l_{2})} ⋯ x_{i_{p},j_{p}}^{(l_{p})} ⪯ x_{r+1,α_{p}}^{(l)}x_{r+2,α_{p−1}}^{(l)} ⋯ x_{p,α_{r+1}}^{(l)}x_{1,α_{r}}^{(l+1)} ⋯ x_{r,α_{1}}^{(l+1)}. (1)

We may assume x_{i_{1},j_{1}}^{(l_{1})} ≺ x_{i_{2},j_{2}}^{(l_{2})} ≺ ⋯ ≺ x_{i_{p},j_{p}}^{(l_{p})} and hence l_{1} ≤ ⋯ ≤ l_{p}. Since l_{1} + ⋯ + l_{p} = a, either l_{1} < q, from which (1) follows, or else l_{1} = ⋯ = l_{p−r} = l and l_{p+1−r} = ⋯ = l_{p} = l + 1. In the second case, as {i_{1}, …, i_{p}} = {1, …, p} and the monomial is in order, we must have i_{1} < ⋯ < i_{p−r} and i_{p+1−r} < ⋯ < i_{p}. If i_{1} ≤ r, then (1) follows, and if i_{1} = r + 1, then the ordered sequence i_{1}, i_{2}, …, i_{p} equals r+1, r+2, …, p, 1, …, r. Among all monomials satisfying this new second case, the largest in the degree reverse lexicographic order ≺ has the second lower index appearing in reverse order. This completes the proof.

We next introduce some combinatorics to help understand the poset C_{p,m}. A row with shift a consists of p consecutive empty unit boxes shifted a units to the left of a given vertical line. A skew shape is an array of such rows whose shifts are weakly increasing read from top to bottom. For example, the unshaded boxes in the figure on the left form a skew shape (with shifts 1,2,2, and 5) while those in the other figure do not:

A (skew) tableau T is a filling of a skew shape with integers that increase across each row. When the entries lie in [m + p], the ith row of a tableau is a sequence α_{(i)} ∈ \binom{[m+p]}{p}. If a_{i} is the shift of the ith row and T has j rows, then T corresponds

to a monomial α_{(1)}^{(a_{1})}α_{(2)}^{(a_{2})} ⋯ α_{(j)}^{(a_{j})}. Conversely, any monomial in the variables α^{(a)} corresponds to a tableau. A tableau T is standard if the entries are weakly increasing in each column, read top to bottom. Equivalently, T is standard if we have α_{(1)}^{(a_{1})} ≤

α_{(2)}^{(a_{2})} ≤ ⋯ ≤ α_{(j)}^{(a_{j})} in C_{p,m}. For example, the following two tableaux correspond to the monomials 345^{(0)}123^{(1)}245^{(3)} and 135^{(0)}123^{(1)}257^{(3)}. The first tableau is not standard and the second tableau is standard:

The elements of the poset C_{p,m}^{q} are represented by one-row tableaux with entries in [m + p] and shift at most q. Two elements satisfy α^{(a)} ≤ β^{(b)} if and only if the two-rowed tableau T = α^{(a)}β^{(b)} is standard. This representation implies that C_{p,m}^{q} is a distributive lattice. Indeed, the two lattice operations ∧ and ∨ are described as follows. If a two-rowed tableau T = α^{(a)}β^{(b)} is non-standard then interchanging the entries in every column in which a violation (α_{a−b+i} < β_{i}) occurs yields a standard tableau. The first row of this new tableau is the meet α^{(a)} ∧ β^{(b)} of α^{(a)} and β^{(b)} in C_{p,m}^{q} and the second row is their join α^{(a)} ∨ β^{(b)}.

Let ψ : k[C_{p,m}^{q}] → k[X] denote the k-algebra homomorphism which sends the variable α^{(a)} to the monomial in_{≺}(φ(α^{(a)})). Its kernel is a toric ideal (i.e. binomial prime) in k[C_{p,m}^{q}].

Proposition 4. The reduced Gröbner basis for the kernel of ψ consists of the binomials

\underline{α^{(a)} · β^{(b)}} − (α^{(a)} ∨ β^{(b)}) · (α^{(a)} ∧ β^{(b)}),

where α^{(a)}, β^{(b)} runs over all incomparable pairs of C_{p,m}^{q}. The initial monomial is underlined.

Proof. This follows from Hibi’s Theorem [9] since C_{p,m}^{q} is a distributive lattice.

Corollary 5. The set of standard tableaux is a k-basis for k[C_{p,m}^{q}]/kernel(ψ) = image(ψ).

Here is a typical element in the reduced Gröbner basis of kernel(ψ) for p = 5, m = 4, q = 9:

\underline{45789^{(1)} · 12356^{(3)}} − 35689^{(1)} · 12457^{(3)}.

Note that the second monomial corresponds to a standard tableau while the first does not.

We write T_{p,m}^{q} for the projective toric variety cut out by the binomials in Lemma 4. Its coordinate ring is the subalgebra image(ψ) of k[X]. The geometry of toric varieties associated with distributive lattices is discussed in [22]. The analogue to Corollary 5 always holds, i.e., multichains in the poset correspond to basis monomials in the coordinate ring.

Corollary 6. The degree of the toric variety T_{p,m}^{q} is the number of maximal chains in C_{p,m}^{q}.

The closed intervals of the poset C_{p,m}^{q} are also distributive lattices. They are denoted

[β^{(b)}, α^{(a)}] := {γ^{(c)} ∈ C_{p,m}^{q} : β^{(b)} ≤ γ^{(c)} ≤ α^{(a)}}.

Proposition 4 and Corollaries 5 and 6 hold essentially verbatim for the distributive sublattice [β^{(b)}, α^{(a)}] as well. The projective toric variety associated with [β^{(b)}, α^{(a)}] is gotten from the toric variety of C_{p,m}^{q} by setting γ^{(c)} = 0 for all γ^{(c)} ∉ [β^{(b)}, α^{(a)}]. The degree of that variety is the number of saturated chains in C_{p,m}^{q} which start at β^{(b)} and end at α^{(a)}.

We close this section with an alternative proof, to be used in Section 4, for the fact that C_{p,m} is a distributive lattice. We claim that C_{p,m} is a sublattice of Young’s lattice. Given α^{(a)} ∈ C_{p,m}, write a = pl + r with integers p > r ≥ 0, and define a sequence J(α^{(a)}) by

J(α^{(a)})_{i} := \{ \begin{matrix}l(m + p) + α_{r+i} & if 1 ≤ i ≤ p − r \\ (l + 1)(m + p) + α_{i−p+r} & if p − r < i ≤ p \end{matrix} . (2)

This gives an order-preserving bijection between the poset C_{p,m} and the poset of sequences J := j_{1} < j_{2} < ⋯ < j_{p} of positive integers with j_{p} − (m + p) < j_{1}, and it preserves meet and join. This bijection preserves the rank function in the two distributive lattices:

|α^{(a)}| := a(m + p) + ∑_{j=1}^{p}(α_{j} − j) = ∑_{i=1}^{p}(J(α^{(a)})_{i} − i ) =: |J(α^{(a)})|. (3)

3. The quantum Grassmannian

Let Grass_{p}k^{m+p} denote the Grassmannian of p-planes in the vector space k^{m+p}. This is a smooth projective variety of dimension mp. Consider the space S_{p,m}^{q} of maps ℙ^{1} → Grass_{p}k^{m+p} of degree q. Such a map may be (non-uniquely) represented as the row space of a p × (m + p)-matrix of polynomials in t whose maximal minors have degree q. Results in [4] imply that it suffices to consider the matrices M(t) in the introduction. The coefficients of these maximal minors define the Plücker embedding of S_{p,m}^{q} into ℙ(∧^{p}k^{m+p} ⊗ k^{q+1}); see [18, 16]. The quantum Grassmannian K_{p,m}^{q} is the Zariski closure of S_{p,m}^{q} in this Plücker embedding. It is an irreducible projective variety of dimension mp + q(m + p). Its prime ideal is kernel(φ) ⊂ k[C_{p,m}^{q}] and its coordinate ring is our subalgebra image(φ) ⊂ k[X].

The quantum Grassmannian K_{p,m}^{q} is singular and it differs from other spaces used to study rational curves in Grassmann varieties (the quot scheme [18], the Kontsevich space of stable maps [12], or the set of autoregressive systems [14]). Nevertheless, K_{p,m}^{q} has been crucial in two important advances: in computing the intersection number degree(K_{p,m}^{q}) in quantum cohomology [15], and in showing that this intersection problem can be fully solved over the real numbers [17]. Our result will give a new derivation of this intersection number.

Corollary 7. [15] The degree of K_{p,m}^{q} is the number of maximal chains in C_{p,m}^{q}.

Proof. This follows immediately from Theorem 2 and Corollary 6.

This degree can also be computed in the small quantum cohomology ring of the Grassmannian [2]. Note that degree(K_{2,3}^{1}) = 55, by counting maximal chains in Figure 1. Ravi, Rosenthal, and Wang [15] were motivated by a problem in applied mathematics. The degree of K_{p,m}^{q} is the number of dynamic feedback compensators that stabilize a certain linear system, in the sense of systems theory. This number can be described in classical projective geometry as follows. The Schubert subvariety of Grass_{p}k^{m+p} consisting of p-planes meeting a fixed m-plane L is a hyperplane section in the Plücker embedding of Grass_{p}k^{m+p}. Thus the set of maps M ∈ S_{p,m}^{q} such that M(t) meets L non-trivially is a hyperplane section of S_{p,m}^{q} in its Plücker embedding. Since GL_{m+p}(k) acts transitively on Grass_{p}k^{m+p}, Kleiman’s Theorem on generic transversality implies the following statement when k is algebraically closed of characteristic zero. Set N := mp + q(m + p) and suppose t_{1}, …, t_{N} ∈ ℙ^{1} are general points and L_{1}, …, L_{N} are general m-planes in k^{m+p}, then the degree of K_{p,m}^{q} counts those maps M for which M(t_{i}) meets L_{i} non-trivially for each i = 1, …, N. As to computing the desired maps M numerically, we note that the sagbi basis in Theorem 1 and the Gröbner basis in Theorem 2 each lead to an optimal homotopy algorithm for finding these degree(K_{p,m}^{q}) maps. These algorithms generalize the ones in [11].

Remark 8. The sagbi basis in Theorem 1 defines a flat deformation from the quantum Grassmannian K_{p,m}^{q} to the projective toric variety T_{p,m}^{q} associated with the poset C_{p,m}^{q}.

See [5] for a precise algebraic discussion of such deformations, and see [20, Equation (11.9)] for the simplest example relevant to us, namely, K_{2,3}^{0} = Grass_{2}k^{5}. The flat deformation is given algebraically by deleting all but the first two terms in the Gröbner basis elements S(γ^{(c)}, δ^{(d)}) given in Theorem 2. Consider the deformation from K_{3,3}^{3} to T_{3,3}^{3}. The incomparable pair 156^{(1)} and 234^{(2)} in C_{3,3}^{3} indexes the quadratic polynomial S(156^{(1)}, 234^{(2)}) =

\begin{matrix}\underline{156^{(1)}234^{(2)} − 146^{(1)}235^{(2)}} + 145^{(1)}236^{(2)} + 136^{(1)}245^{(2)} − 135^{(1)}246^{(2)} \\ +134^{(1)}256^{(2)} − 126^{(1)}345^{(2)} + 125^{(1)}346^{(2)} − 124^{(1)}356^{(2)} + 123^{(1)}456^{(2)} \\ −456^{(0)}123^{(3)} + 356^{(0)}124^{(3)} − 346^{(0)}125^{(3)} + 345^{(0)}126^{(3)} − 256^{(0)}134^{(3)} \\ +246^{(0)}135^{(3)} − 245^{(0)}136^{(3)} − 236^{(0)}145^{(3)} + 235^{(0)}146^{(3)} − 234^{(0)}156^{(3)} \\ +2·156^{(0)}234^{(3)} − 2·146^{(0)}235^{(3)} + 2·145^{(0)}236^{(3)} + 2·136^{(0)}245^{(3)} − 2·135^{(0)}246^{(3)} \\ +2·134^{(0)}256^{(3)} − 2·126^{(0)}345^{(3)} + 2·125^{(0)}346^{(3)} − 2·124^{(0)}356^{(3)} + 2·123^{(0)}456^{(3)}, \end{matrix} (4)

which vanishes on the quantum Grassmannian K_{3,3}^{3}. The underlined leading binomial vanishes on the toric variety T_{3,3}^{3}, by Proposition 4. Our main technical problem, to be solved in the next section, is the reconstruction of quadrics such as (4) from their leading binomial.

A key tool in proving Theorems 1 and 2 is the Schubert decomposition of the quantum Grassmannian K_{p,m}^{q} indexed by C_{p,m}^{q}. For α^{(a)} ∈ C_{p,m}^{q}, the quantum Schubert variety is

Z_{α^{(a)}} := { (γ^{(c)}) ∈ K_{p,m}^{q} | γ^{(c)} = 0 if γ^{(c)} ≰ α^{(a)}} .

More generally, for β^{(b)} ≤ α^{(a)} in C_{p,m}^{q}, we define the skew quantum Schubert variety

Z_{α^{(a)}/β^{(b)}} := { (γ^{(c)}) ∈ K_{p,m}^{q} | γ^{(c)} = 0 if γ^{(c)} ∉ [β^{(b)}, α^{(a)}]} .

Among the quantum Schubert varieties of K_{p,m}^{q} are the K_{p,m}^{d} for d < q; namely, if δ^{(d)} is the supremum of C_{p,m}^{d}, then K_{p,m}^{d} = Z_{δ^{(d)}}. This allows us to deduce assertions about the general quantum Grassmannian K_{p,m}^{q} from results about quantum Schubert varieties of K_{p,m}^{pn}.

The quantum Schubert varieties and skew quantum Schubert varieties have rational parameterizations which are constructed as follows. Let α^{(a)} ∈ C_{p,m}^{pn} and write a = ps+r with integers p > r ≥ 0. We define the matrix M_{α^{(a)}}(t) to be the specialization of M(t) where

x_{i,j}^{(l)} = 0 if \{ \begin{matrix}(l > s + 1 and i ≤ r) & or (l = s + 1 and j > α_{r+1−i}) or \\ (l > s and i > r) & or (l = s and j > α_{p+r+1−i}) . \end{matrix}

Here we use the conventions α_{ν} = 0 if ν ≤ 0 and α_{ν} = +∞ if ν > p. For example,

M_{235^{(2)}}(t) = [ \begin{matrix}x_{1,1}^{(0)} + x_{1,1}^{(1)} · t & x_{1,2}^{(0)} + x_{1,2}^{(1)} · t & x_{1,3}^{(0)} + x_{1,3}^{(1)} · t & x_{1,4}^{(0)} & x_{1,5}^{(0)} & x_{1,6}^{(0)} \\ x_{2,1}^{(0)} + x_{2,1}^{(1)} · t & x_{2,2}^{(0)} + x_{2,2}^{(1)} · t & x_{2,3}^{(0)} & x_{2,4}^{(0)} & x_{2,5}^{(0)} & x_{2,6}^{(0)} \\ x_{3,1}^{(0)} & x_{3,2}^{(0)} & x_{3,3}^{(0)} & x_{3,4}^{(0)} & x_{3,5}^{(0)} & 0 \end{matrix} ] .

If we specialize the variables x_{i,j}^{(l)} in M_{α^{(a)}}(t) to field elements in k in such a way that the resulting matrix over k(t) has maximal row rank, then that matrix defines a map from k to Grass_{p}k^{m+p}. If we extend this to ℙ^{1}, we obtain a map in Z_{α^{(a)}}.

Proposition 9 below implies that such maps constitute a dense subset of Z_{α^{(a)}}. This means that the coefficients with respect to t of the maximal minors of M_{α^{(a)}}(t) give a rational parameterization of Z_{α^{(a)}}.

This construction extends to skew quantum Schubert varieties as follows. Given β^{(b)} ≤ α^{(a)}, write b = ps+r with integers p > r ≥ 0 and define the matrix M_{α^{(a)}/β^{(b)}}(t) to be the specialization of M_{α^{(a)}}(t) where

x_{i,j}^{(l)} = 0 if \{ \begin{matrix}(l < s + 1 and i ≤ r) & or (l = s + 1 and j < β_{r+1−i}) or \\ (l < s and i > r) & or (l = s and j < β_{p+r+1−i}) \end{matrix} .

The matrix M_{α^{(a)}/β^{(b)}}(t) gives a rational map into Z_{α^{(a)}/β^{(b)}}, which is described algebraically as follows. We define φ_{α^{(a)}} and φ_{α^{(a)}/β^{(b)}} to be the composition of the map φ : k[C_{p,m}^{np}] → k[X] with the specializations to M_{α^{(a)}}(t) and M_{α^{(a)}/β^{(b)}}(t) respectively. We claim that these matrices parameterize dense subsets of the (skew) quantum Schubert varieties.

Proposition 9. The kernel of φ_{α^{(a)}} is the homogeneous ideal of the quantum Schubert variety Z_{α^{(a)}}. Likewise, the kernel of φ_{α^{(a)}/β^{(b)}} is the homogeneous ideal of the skew quantum Schubert variety Z_{α^{(a)}/β^{(b)}}. In particular, the varieties Z_{α^{(a)}} and Z_{α^{(a)}/β^{(b)}} are irreducible.

We postpone the proof of this proposition until the next section. Here is an example which illustrates the parameterization of skew quantum Schubert varieties for p = m = 3:

M_{235^{(2)}/146^{(1)}}(t) = [ \begin{matrix}x_{1,1}^{(1)} · t & x_{1,2}^{(1)} · t & x_{1,3}^{(1)} · t & 0 & 0 & 0 \\ x_{2,1}^{(1)} · t & x_{2,2}^{(1)} · t & 0 & 0 & 0 & x_{2,6}^{(0)} \\ 0 & 0 & 0 & x_{3,4}^{(0)} & x_{3,5}^{(0)} & 0 \end{matrix} ] .

We evaluate the 3 × 3-minors of this matrix to find the k-algebra homomorphism φ_{235^{(2)}/146^{(1)}}. It takes polynomials in 12 variables γ^{(c)} to polynomials in 8 variables x_{i,j}^{(l)} as follows:

146^{(1)} ↦ −x_{1,1}^{(1)}x_{2,6}^{(0)}x_{3,4}^{(0)} , 156^{(1)} ↦ −x_{1,1}^{(1)}x_{2,6}^{(0)}x_{3,5}^{(0)} , 246^{(1)} ↦ −x_{1,2}^{(1)}x_{2,6}^{(0)}x_{3,4}^{(0)} ,
256^{(1)} ↦ −x_{1,2}^{(1)}x_{2,6}^{(0)}x_{3,5}^{(0)} , 346^{(1)} ↦ −x_{1,3}^{(1)}x_{2,6}^{(0)}x_{3,4}^{(0)} , 356^{(1)} ↦ −x_{1,3}^{(1)}x_{2,6}^{(0)}x_{3,5}^{(0)} ,
124^{(2)} ↦ x_{1,1}^{(1)}x_{2,2}^{(1)}x_{3,4}^{(0)} − x_{2,1}^{(1)}x_{1,2}^{(1)}x_{3,4}^{(0)} , 125^{(2)} ↦ x_{1,1}^{(1)}x_{2,2}^{(1)}x_{3,5}^{(0)} − x_{2,1}^{(1)}x_{1,2}^{(1)}x_{3,5}^{(0)} ,
134^{(2)} ↦ −x_{2,1}^{(1)}x_{1,3}^{(1)}x_{3,4}^{(0)} , 135^{(2)} ↦ −x_{2,1}^{(1)}x_{1,3}^{(1)}x_{3,5}^{(0)} ,
234^{(2)} ↦ −x_{2,2}^{(1)}x_{1,3}^{(1)}x_{3,4}^{(0)} , 235^{(2)} ↦ −x_{2,2}^{(1)}x_{1,3}^{(1)}x_{3,5}^{(0)}.

The 12 variables γ^{(c)} appearing on the left sides above are precisely the elements in the interval [146^{(1)}, 235^{(2)}] of the distributive lattice C_{3,3}. There are 18 incomparable pairs in this interval, each giving a quadratic generator for the kernel of φ_{235^{(2)}/146^{(1)}}. This set of 18 quadrics consists of 14 binomials and four trinomials, and it equals the reduced Gröbner basis with respect to ≺. For example, one of the 14 binomials in this Gröbner basis is the underlined leading binomial of S(156^{(1)}, 234^{(2)}) in (4), and one of the four trinomials is

\underline{346^{(1)} · 125^{(2)}} − 246^{(1)} · 135^{(2)} + 146^{(1)} · 235^{(2)}.

The underlined term is an incomparable pair in [146^{(1)}, 235^{(2)}], while the other two monomials are comparable pairs. Erasing the third term gives a binomial as in Proposition 4.

4. Construction of Straightening Syzygies

The following theorem is the technical heart of this paper. All three of Theorem 1, Theorem 2, and Proposition 9 will be derived from Theorem 10 in the end of this section.

Theorem 10. Let γ^{(c)}, δ^{(d)} be a pair of incomparable variables in the poset C_{p,m}^{np}. There is a quadric S(γ^{(c)}, δ^{(d)}) in the kernel of φ : k[C_{p,m}^{np}] → k[X] whose first two monomials are

γ^{(c)} · δ^{(d)} − (γ^{(c)} ∨ δ^{(d)}) · (γ^{(c)} ∧ δ^{(d)}).

Moreover, if λβ^{(b)}α^{(a)} is any non-initial monomial in S(γ^{(c)}, δ^{(d)}), then γ^{(c)}, δ^{(d)} ∈ [β^{(b)}, α^{(a)}].

The pair β^{(b)}α^{(a)} in the second assertion is necessarily standard, i.e. β^{(b)} < α^{(a)}. The quadrics S(γ^{(c)}, δ^{(d)}) are not constructed explicitly, but rather through an iterative procedure modeled on the subduction algorithm in image(φ). A main idea is to utilize the well-known subduction process [19, Algorithm 3.2.6] modulo the p × p-minors of a generic p × N-matrix.

Set N := (n + 1)(m + p). Let N be the p × N-matrix whose i, jth entry is x_{i,r}^{(l)}, where j = (m + p)l + r with 1 ≤ r ≤ m + p. If N_{l} is the submatrix of N consisting of the entries x_{i,j}^{(l)}, then N is the concatenation of N_{0}, N_{1}, …, N_{n} and M(t) = N_{0} + tN_{1} + ⋯ + t^{n}N_{n}. Sequences J : j_{1} < ⋯ < j_{p} ∈ \binom{[N]}{p} are regarded as variables. We write ϕ(J) for the Jth maximal minor of N. Young’s poset on sequences J is given by componentwise comparison and is graded via |J| := ∑_{i}(j_{i} − i). The coefficient φ(α^{(a)}) of t^{a} in the αth maximal minor of M(t) is an alternating sum of maximal minors of N. The exact formula is

φ(α^{(a)}) = ∑_{|J| = |α^{(a)}|_{J≡α mod(m+p)}}ϵ_{J} · ϕ(J), (5)

where ϵ_{J} is the sign of the permutation that orders the following sequence:

j_{1} mod (m + p), j_{2} mod (m + p), …, j_{p} mod (m + p).

The polynomial rings k[C_{p,m}^{np}] and k[\binom{[N]}{p}] are graded with deg α^{(a)} = |α^{(a)}| and deg J = |J|. Consider the degree-preserving k-algebra homomorphism π : k[C_{p,m}^{np}] → k[\binom{[N]}{p}] defined by

π(α^{(a)}) = ∑_{|J| = |α^{(a)}|_{J≡α mod(m+p)}}ϵ_{J} · J . (6)

Lexicographic order on the sequences J ∈ \binom{[N]}{p} gives a linear extension of Young’s poset. In this ordering, the initial term of (6) is the sequence J(α^{(a)}) defined in (2). This sequence is characterized by in_{≺} φ(α^{(a)}) = in_{≺} ϕ(J(α^{(a)})). It can be checked that all other terms ϵ_{J} ·J appearing in (6) satisfy J_{1} < J(α^{(a)})_{1} and J_{p}−J_{1} > m+p. For example, for m = 4,

π(235^{(2)}) = \underline{(5, 9, 10)} − (3, 9, 12) + (3, 5, 16) + (2, 10, 12) − (2, 5, 17) + (2, 3, 19),

and in_{≺}(φ( 235^{(2)} )) = in_{≺}(ϕ(5, 9, 10)) = x_{3,5}^{(0)}x_{1,3}^{(1)}x_{2,2}^{(1)}.

For J ∈ \binom{[N]}{p}, let N_{J} be the specialization of N where in each row i, all entries in columns greater than j_{i} are set to zero. Under the identification of N with M(t), we have N_{J(α^{(a)})} = M_{α^{(a)}}. Let ϕ_{J} : k[\binom{[N]}{p}] → k[X] denote the k-algebra homomorphism which maps the formal variable I to the Ith maximal minor of N_{J}. Then ϕ_{J}(I) vanishes unless I ≤ J. In particular, if |I| = |J|, then ϕ_{J}(I) vanishes unless I = J, and in that case, it is just the product of the last non-zero variables in each row of N_{J}. From this it follows that

\begin{matrix}φ_{α^{(a)}} = ϕ_{J(α^{(a)})} ∘ π \\ φ_{α^{(a)}}(α^{(a)}) = ϕ_{J(α^{(a)})}(J(α^{(a)})) = in_{≺} φ(α^{(a)}) = ψ(α^{(a)}). \end{matrix} (7)

In the Plücker embedding of Grass_{p}k^{N} into ℙ(∧^{p}k^{N}), the Schubert variety indexed by J is

Ω_{J} := {y = (y_{I}) ∈ Grass_{p}k^{N} | y_{I} = 0 if I ≰ J} .

The homogeneous ideal I(Ω_{J}) which defines this Schubert variety is precisely the kernel of ϕ_{J}. The following identity of ideals in k[\binom{[N]}{p}] follows from the classical Plücker relations:

Proposition 11. For any J ∈ \binom{[N]}{p} we have

⋂_{I<J} I(Ω_{I}) = I(Ω_{J}) + ⟨ J ⟩ .

The map π : k[C_{p,m}^{np}] → k[\binom{[N]}{p}] induces a birational isomorphism π^{∗} : Grass_{p}k^{N} −−→ K_{p,m}^{np}. From the identification of M_{α^{(a)}}(t) with N_{J(α^{(a)})} and Proposition 9, we will see that π^{∗}(Ω_{J(α^{(a)})}) is a dense subset of Z_{α^{(a)}}. We also consider the image under π^{∗} of the Schubert varieties Ω_{J} for J < J(α^{(a)}).

Proposition 12 (Ravi-Rosenthal-Wang [15]). If J < J(α^{(a)}), then

π^{∗}(Ω_{J}) ⊂ ⋃_{β^{(b)}<α^{(a)}} Z_{β^{(b)}} .

Proof. The inclusion Ω_{J} ⊂ Ω_{J(α^{(a)})} implies π^{∗}(Ω_{J}) ⊂ Z_{α^{(a)}}. Since φ_{α^{(a)}}(α^{(a)}) is the product of leading entries in the rows of N_{J(α^{(a)})}, it follows that φ_{α^{(a)}}(α^{(a)}) vanishes under the specialization to N_{J}, and hence π(α^{(a)}) vanishes on Ω_{J}. This implies our claim because ⋃_{β^{(b)}<α^{(a)}} Z_{β^{(b)}} is defined as a subvariety of Z_{β^{(b)}} by the vanishing of α^{(a)}.

For L < J in Young’s poset, define N_{J/L} to be the specialization of N where in the ith row, only the entries in columns l_{i}, l_{i} + 1, …, j_{i} are non-zero. Then M_{α^{(a)}/β^{(b)}}(t) is the specialization of M(t) corresponding to N_{J(α^{(a)})/J(β^{(b)})}(t). Define the k-algebra homomorphism ϕ_{J/I} : k[\binom{[N]}{p}] → k[X] by evaluating the appropriate minors on N_{J/L}. We observe that

\begin{aligned}φ_{α^{(a)}/β^{(b)}}(α^{(a)}) & = & in_{≺} φ(α^{(a)}) = ϕ_{J(α^{(a)})/J(β^{(b)})}(J(α^{(a)})), \\ φ_{α^{(a)}/β^{(b)}}(β^{(b)}) & = & in_{≺} φ(β^{(b)}) = ϕ_{J(α^{(a)})/J(β^{(b)})}(J(β^{(b)})). \end{aligned} (8)

The following lemma is very useful in our proof of Theorem 10.

Lemma 13. Fix α^{(a)} ∈ C_{p,m}^{np} and let f ∈ k[C_{p,m}^{np}] be a quadratic form of degree d.

1. Suppose that φ_{β^{(b)}}(f) = 0 for all β^{(b)} < α^{(a)}. Then there exist constants λ_{J} ∈ k

with

φ_{α^{(a)}}(f) = φ_{α^{(a)}}(α^{(a)}) · ∑_{J ∈ \binom{[N]}{p}_{|J|+|α^{(a)}|=d}} λ_{J} · ϕ_{J(α^{(a)})}(J) .

2. Suppose β^{(b)} < α^{(a)} and φ_{α^{(a)}/γ^{(c)}}(f) = 0 for all β^{(b)} < γ^{(c)} ≤ α^{(a)}. For some λ_{J} ∈ k,

φ_{α^{(a)}/β^{(b)}}(f) = φ_{α^{(a)}/β^{(b)}}(β^{(b)}) · ∑_{J ∈ \binom{[N]}{p}_{|J|+|β^{(b)}|=d}} λ_{J} · ϕ_{J(α^{(a)})/J(β^{(b)})}(J) .

Proof. We only prove part 1. The hypothesis states that ϕ_{J(α^{(a)})}(π(f)) = φ_{α^{(a)}}(f) vanishes on all matrices N_{J(β^{(b)})} for β^{(b)} < α^{(a)}. Proposition 12 implies that π(f) vanishes on all Schubert varieties Ω_{J} with J < J(α^{(a)}). But then, using Proposition 11,

π(f) ∈ ⋂_{J<J(α^{(a)})} I(Ω_{J}) = I(Ω_{J(α^{(a)})}) + ⟨ J(α^{(a)}) ⟩ .

This means π(f) = g + J(α^{(a)}) · h, where g ∈ I(Ω_{J(α^{(a)})}) = ker(ϕ_{J(α^{(a)})}) and h ∈ k[\binom{[N]}{p}] is a linear form of degree d − |α^{(a)}|. Such a linear form can be written as follows

h = ∑_{|J|+|α^{(a)}|=d}λ_{J} J .

By applying the map ϕ_{J(α^{(a)})} to both sides of the equation π(f) = g + J(α^{(a)}) · h, we obtain the first assertion of Lemma 13. Part 2 is proved by similar arguments.

Our proof of Theorem 10 will show that the sums in Lemma 13 are actually sums of terms of the form λ_{J(δ^{(d)})} · φ_{α^{(a)}}(δ^{(d)}) and λ_{J(δ^{(d)})} · φ_{α^{(a)}/β^{(b)}}(δ^{(d)}) respectively. The next lemma provides the initial step in our inductive proof of Theorem 10.

Lemma 14. Let γ^{(c)} and δ^{(d)} be incomparable variables in the poset C_{p,m}^{np} and set α^{(a)} := γ^{(c)} ∨ δ^{(d)}. Then φ_{α^{(a)}}(γ^{(c)} · δ^{(d)} − γ^{(c)} ∨ δ^{(d)} · γ^{(c)} ∧ δ^{(d)}) = 0.

Proof. We prove the lemma by inductively showing that, for each β^{(b)} ≤ α^{(a)},

φ_{α^{(a)}/β^{(b)}}( γ^{(c)} · δ^{(d)} − γ^{(c)} ∨ δ^{(d)} · γ^{(c)} ∧ δ^{(d)} ) = 0. (9)

If β^{(b)} ≰ γ^{(c)} ∧ δ^{(d)}, then φ_{α^{(a)}/β^{(b)}}(γ^{(c)} ∧ δ^{(d)}) vanishes, and either φ_{α^{(a)}/β^{(b)}}(γ^{(c)}) vanishes or φ_{α^{(a)}/β^{(b)}}(δ^{(d)}) vanishes. This implies that (9) holds.

Next suppose β^{(b)} = γ^{(c)} ∧ δ^{(d)}. We claim that φ_{α^{(a)}/β^{(b)}} maps each variable appearing in (9) to its initial term in k[X]. In view of Proposition 4, this claim implies (9). To establish this claim, we need only show that φ_{α^{(a)}/β^{(b)}}(γ^{(c)}) = in_{≺}φ(γ^{(c)}), as the case for δ^{(d)} is similar and that of the other terms follow from (8). Consider the expansion of φ_{α^{(a)}/β^{(b)}}(γ^{(c)}) in terms of the minors ϕ(J) of N_{J(α^{(a)})/J(β^{(b)})}. First observe that the submatrix given by the columns from J(γ^{(c)}) is block anti-diagonal, with each block either upper or lower triangular along its anti-diagonal. This is because for each i, J(γ^{(c)})_{i} is either J(β^{(b)})_{i} or J(α^{(a)})_{i}, and the non-zero entries in the ith row of N_{J(α^{(a)})/J(β^{(b)})} lie between these two numbers. Thus the contribution of term J(γ^{(c)}) to φ_{α^{(a)}/β^{(b)}}(γ^{(c)}) is simply in_{≺} φ(γ^{(c)}).

We claim there are no other terms. If there were another term indexed by L, then the Lth maximal minor of N_{J(α^{(a)})/J(β^{(b)})} would be non-zero, and so J(β^{(b)}) ≤ L ≤ J(α^{(a)}). Thus

J(β^{(b)}) = J(γ^{(c)}) ∧ J(δ^{(d)}) ≤ L ∧ J(δ^{(d)}).

Comparing the first components of these sequences gives min{J(γ^{(c)})_{1}, J(δ^{(d)})_{1}} ≤ L_{1}. Since L_{1} < J(γ^{(c)})_{1}, this implies J(δ^{(d)})_{1} ≤ L_{1}. Similarly, using J(α^{(a)}) ≥ L∨J(δ^{(d)}), we see that J(δ^{(d)})_{p} ≥ L_{p}. Lastly, as L is a summand in π(γ^{(c)}) and L ≠ J(γ^{(c)}), we have L_{p} − L_{1} > m + p and thus

m + p ≥ J(δ^{(d)})_{p} − J(δ^{(d)})_{1} ≥ L_{p} − L_{1} > m + p,

a contradiction, which proves the claim. Thus (9) holds for β^{(b)} = γ^{(c)} ∧ δ^{(d)}.

Finally, let ζ^{(z)} < γ^{(c)} ∧ δ^{(d)} and suppose that (9) holds for all β^{(b)} with ζ^{(z)} < β^{(b)} ≤ α^{(a)}. Then by Lemma 13,

φ_{α^{(a)}/ζ^{(z)}}(γ^{(c)}·δ^{(d)} − γ^{(c)}∨δ^{(d)}·γ^{(c)}∧δ^{(d)}) = φ_{α^{(a)}/ζ^{(z)}}(ζ^{(z)})·∑_{J} λ_{J}·ϕ_{J(α^{(a)})/J(ζ^{(z)})}(J) ,

the sum over sequences J of rank |J| = |γ^{(c)}| + |δ^{(d)}| − |ζ^{(z)}|. But this exceeds the rank of α^{(a)}, since ζ^{(z)} < γ^{(c)} ∧ δ^{(d)} and |α^{(a)}| + |γ^{(c)} ∧ δ^{(d)}| = |γ^{(c)}| + |δ^{(d)}|. Thus the sum vanishes and so (9) holds for all β^{(b)} ≤ α^{(a)}, which proves the lemma.

Proof of Theorem 10. Let γ^{(c)} and δ^{(d)} be incomparable variables in the poset C_{p,m}^{np}. For each α^{(a)} ∈ C_{p,m}^{np} we inductively construct quadratic polynomials S_{α^{(a)}}(γ^{(c)}, δ^{(d)}) ∈ k[β^{(s)} | β^{(s)} ≤ α^{(a)}], and then show S_{α^{(a)}}(γ^{(c)}, δ^{(d)}) is in the kernel of the map φ_{α^{(a)}}. The case when α^{(a)} is the top element in the poset C_{p,m}^{np} proves the theorem. These polynomials have the following restriction property: If β^{(b)} < α^{(a)}, then S_{β^{(b)}}(γ^{(c)}, δ^{(d)}) is the image of S_{α^{(a)}}(γ^{(c)}, δ^{(d)}) under the map which sets variables ζ^{(z)} ≰ β^{(b)} to zero. They also have the further homogeneity properties that each non-zero term λζ^{(z)}β^{(b)} must have z + b = c + d and satisfy the multiset equality β ∪ ζ = γ ∪ δ, and if it is not the initial term, then γ^{(c)}, δ^{(d)} ∈ [ζ^{(z)}, β^{(b)}].

For α^{(a)} ≱ γ^{(c)} ∨ δ^{(d)}, set S_{α^{(a)}}(γ^{(c)}, δ^{(d)}) := 0 and if α^{(a)} = γ^{(c)} ∨ δ^{(d)}, then set

S_{α^{(a)}}(γ^{(c)}, δ^{(d)}) := γ^{(c)} · δ^{(d)} − γ^{(c)} ∨ δ^{(d)} · γ^{(c)} ∧ δ^{(d)}.

These polynomials have the restriction and homogeneity properties, and, for α^{(a)} ≯ γ^{(c)} ∨ δ^{(d)}, we have φ_{α^{(a)}}(S_{α^{(a)}}(γ^{(c)}, δ^{(d)})) = 0, by Lemma 14.

Let α^{(a)} > γ^{(c)} ∨ δ^{(d)} and suppose we have constructed S_{β^{(b)}}(γ^{(c)}, δ^{(d)}) for each β^{(b)} < α^{(a)}. By the restriction property, there is a polynomial S′ ∈ k[β^{(b)} | β^{(b)} < α^{(a)}] which restricts to S_{β^{(b)}}(γ^{(c)}, δ^{(d)}) for each β^{(b)} < α^{(a)}. Thus φ_{β^{(b)}}(S′) = 0 for all β^{(b)} < α^{(a)}.

Set e := |γ^{(c)}| + |δ^{(d)}|, the degree of S′. By Lemma 13,

φ_{α^{(a)}}(S′) = φ_{α^{(a)}}(α^{(a)}) · ∑_{|J|+|α^{(a)}|=e} λ_{J} · ϕ_{J(α^{(a)})}(J) . (10)

If we consider the columns of M_{α^{(a)}}(t) involved in φ_{α^{(a)}}(S′), we see that this sum is further restricted to those J which satisfy the multiset equality (γ ∪ δ) \ α ≡ J mod (m+p), with J mod (m+p) consisting of distinct integers, and with J ≤ J(α^{(a)}). If there are no such J, then φ_{α^{(a)}}(S′) = 0 and we set S_{α^{(a)}}(γ^{(c)}, δ^{(d)}) = S′.

Otherwise, let z := c + d − a and ζ := (γ ∪ δ) \ α. Then the summands in (10) are among those J which appear in π(ζ^{(z)}) so we have J(ζ^{(z)}) < J(α^{(a)}) and hence ζ^{(z)} < α^{(a)}. Observe that φ_{α^{(a)}/ζ^{(z)}}(S′) = λ_{J(ζ^{(z)})} · φ_{α^{(a)}/ζ^{(z)}}(α^{(a)}ζ^{(z)}). Define

S_{α^{(a)}}(γ^{(c)}, δ^{(d)}) := S′ − λ_{J(ζ^{(z)})}α^{(a)}ζ^{(z)}.

We claim that if λ_{J(ζ^{(z)})} ≠ 0, then ζ^{(z)} ≤ γ^{(c)}, δ^{(d)}. If not, then every term of S′ contains a variable ξ^{(x)} with ζ^{(z)} ≰ ξ^{(x)}, and so we must have φ_{α^{(a)}/ζ^{(z)}}(S′) = 0, a contradiction.

We complete the proof of Theorem 10 by showing that for β^{(b)} ≤ α^{(a)},

φ_{α^{(a)}/β^{(b)}}(S_{α^{(a)}}(γ^{(c)}, δ^{(d)})) = 0. (11)

If β^{(b)} ≰ ζ^{(z)}, then φ_{α^{(a)}/β^{(b)}}(ζ^{(z)}) = 0 and so φ_{α^{(a)}/β^{(b)}}(S_{α^{(a)}}(γ^{(c)}, δ^{(d)})) = φ_{α^{(a)}/β^{(b)}}(S′), which is zero as ϕ_{J(α^{(a)})/J(β^{(b)})}(J) = 0 for all J which appear in π(ζ^{(z)}). By the construction of S_{α^{(a)}}(γ^{(c)}, δ^{(d)})), we also have φ_{α^{(a)}/ζ^{(z)}}(S_{α^{(a)}}(γ^{(c)}, δ^{(d)})) = 0.

Let ξ^{(x)} < ζ^{(z)} and suppose (11) holds for all β^{(b)} with ξ^{(x)} < β^{(b)}. Then by Lemma 13,

φ_{α^{(a)}/ξ^{(x)}}(S_{α^{(a)}}(γ^{(c)}, δ^{(d)})) = φ_{α^{(a)}/ξ^{(x)}}(ξ^{(x)}) · ∑_{|J|+|ξ^{(x)}|=e} λ_{J} · ϕ_{J(α^{(a)})/J(ξ^{(x)})}(J).

Since |J| = e − |ξ^{(x)}| > e − |ζ^{(z)}| = |α^{(a)}|, each term in the right hand sum is zero.

It is now straightforward to derive all our assertions that were left unproven so far.

Proof of Theorem 1. Theorem 10 together with Proposition 4 shows that the subduction criterion for sagbi bases (see e.g. [5, Proposition 1.1] or [20, Theorem 11.4]) is satisfied.

Proof of Theorem 2. A standard fact on sagbi bases, proved in [5, Corollary 2.2] or in [20, Corollary 11.6 (1)], states that the reduced Gröbner basis for the binomial ideal kernel(ψ) lifts to a reduced Gröbner basis for the non-binomial ideal kernel(φ).

Proof of Proposition 9. If β^{(b)} is the minimal element in the poset C_{p,m}, then φ_{α^{(a)}} and φ_{α^{(a)}/β^{(b)}} have the same kernel, as the varieties Z_{α^{(a)}/β^{(b)}} and Z_{α^{(a)}} are equal. Hence it suffices to prove the second statement about φ_{α^{(a)}/β^{(b)}}. Clearly, the kernel of φ_{α^{(a)}/β^{(b)}} contains the homogeneous ideal of the skew quantum Schubert variety Z_{α^{(a)}/β^{(b)}}. If this containment were proper, then we would also get proper containment at the level of initial ideals with respect to the induced partial term order, which was denoted by A^{T} ω in [20, Chapter 11]. But that is impossible since every binomial relation on the monomials in_{≺} φ_{α^{(a)}/β^{(b)}}(γ^{(c)}) = in_{≺} φ(γ^{(c)}) lifts to a polynomial which vanishes on Z_{α^{(a)}/β^{(b)}}, as shown in the proof Theorem 10.

5. Applications and future directions

We first summarize some algebraic consequences of our main results.

Corollary 15. The coordinate ring of the quantum Grassmannian K_{p,m}^{q} is an algebra with straightening law on the distributive lattice C_{p,m}^{q}. It has a presentation by a noncommutative Gröbner basis consisting of quadratic elements, and, in particular, it is a Koszul algebra.

Proof. The first statement follows from Theorem 1 and the form of the syzygies S(γ^{(c)}, δ^{(d)}) of Theorem 2. For the second statement, consider the coordinate ring of K_{p,m}^{q} as the quotient of the free associative algebra on C_{p,m}^{q} modulo a two-sided ideal. By [7, Proposition 3.2] that two-sided ideal has a quadratic Gröbner basis, obtained from lifting the Gröbner basis in Theorem 2. For the classical Grassmannian (n = 0) this result appeared in [8]. The Koszul property is a well-known consequence of the existence of a quadratic Gröbner basis; see e.g. [8, Theorem 3] for the noncommutative version which is relevant here.

Corollary 16. The coordinate ring of K_{p,m}^{q} is a normal Cohen-Macaulay and Gorenstein domain. It has rational singularities if char(k) = 0 and it is F -rational if char(k) > 0.

Proof. By Corollary 15 and the results of [5], these properties of K_{p,m}^{q} follow from the corresponding properties of the toric variety T_{p,m}^{q}. But these were established in [22], as T_{p,m}^{q} is the toric variety associated to the distributive lattice C_{p,m}^{q}.

We remark that both Corollary 16 and the analog of Corollary 15 (with the poset C_{p,m}^{q} replaced by the appropriate interval) hold for the skew quantum Schubert varieties.

Our next application is the sagbi property of the row-consecutive p × p-minors of a matrix of indeterminates. This result is non-trivial since the set of all p × pminors is not a sagbi basis in general [20, Example 11.3]. A finite sagbi basis for the algebra of all p × p-minors was found by Bruns and Conca [3]. Let L be the p(n + 1) × (m + p)-matrix whose i, jth entry is x_{r,j}^{(l)}, where i = pl + r. This matrix is obtained from N by stacking the matrices N_{0}, …, N_{n}. Let χ : k[C_{p,m}^{np}] → k[X] denote the k-algebra homomorphism which sends the variable α^{(a)} to the αth maximal minor of the submatrix of L consisting of rows a + 1, a + 2, …, a + p. Thus the collection of polynomials χ(α^{(a)}) are the row-consecutive p × p-minors of L.

Theorem 17. The set {χ(α^{(a)}) : α^{(a)} ∈ C_{p,m}^{np}} of row-consecutive p × p-minors of a generic matrix is a sagbi basis with respect to the degree reverse lexicographic term order ≺ on k[X].

Brain Taylor has pointed out this may also be deduced from Proposition 2.7.3 of his Ph.D. Thesis [21].

Proof. Let ω be the weight on the variables in k[X] defined by ω(x_{i,j}^{(l)}) := −(pl+i)^{2}. Then χ(α^{(a)}) = in_{ω}(φ(α^{(a)})), the initial form of φ(α^{(a)}), and we have in_{≺}(χ(α^{(a)})) = in_{≺}(φ(α^{(a)})) for all α^{(a)} ∈ C_{p,m}^{np}. Thus image(φ) and image(χ) have the same initial algebra, and so we deduce the sagbi property for the polynomials χ(α^{(a)}) from Theorem 1.

Let w denote the weight on the variables C_{p,m}^{np} defined by w(α^{(a)}) := −a^{2}. For every incomparable pair γ^{(c)}, δ^{(d)} in the poset C_{p,m}^{np}, we define the quadratic polynomial

R(γ^{(c)}, δ^{(d)}) := in_{w}(S(γ^{(c)}, δ^{(d)})),

where S(γ^{(c)}, δ^{(d)}) is the element of the reduced Gröbner basis for the kernel of φ. For example, R(156^{(1)}, 234^{(2)}) equals the sum of the first ten terms in (4). The weight w is equivalent, modulo the homogeneities of kernel(φ), to the induced weight which was denoted by A^{T} ω in [20, Chapter 11]. The only-if direction in [20, Theorem 11.4] implies

Corollary 18. The reduced Gröbner basis of the kernel of χ consists of the quadratic polynomials R(γ^{(c)}, δ^{(d)}) as γ^{(c)}, δ^{(d)} run over the set of incomparable pairs in the poset C_{p,m}^{np}.

For the Plücker ideal defining the classical Grassmannian (n = 0), an explicit (but non-reduced) quadratic Gröbner basis is known. It appears in the work of Hodge-Pedoe [10] and Doubilet-Rota-Stein [6], and it consists of the van der Waerden syzygies. They are discussed in Gröbner basis language in [19, Section 3.1]. Our next aim is to introduce an analogous non-reduced Gröbner basis for the ideal kernel(φ) of the quantum Grassmannian.

We begin by defining the skew van der Waerden syzygies for its initial ideal

(12) kernel(χ) = in_{w}(kernel(φ)),

which consists of the algebraic relations among the row-consecutive minors. Given a sequence of integers D : 1 ≤ d_{1} < ⋯ < d_{p} ≤ m + p and any integer 0 ≤ a ≤ np, let D^{(a)} denote ±α^{(a)}, where α is the reordering of the sequence D and ± is the sign of the permutation which sorts the sequence D. Let T = α^{(a)}β^{(b)} with a < b be a non-standard tableau and i the smallest index of a violation β_{i} < α_{i−b+a}. Define increasing sequences

\begin{matrix}A := α_{1}, …, α_{i−b+a−1} B := β_{i+1}, …, β_{p} \\ C := β_{1}, …, β_{i}, α_{i−b+a}, …, α_{p}. \end{matrix}

For a subset I ∈ \binom{[p+b−a+1]}{i}, let C_{I} be the corresponding numbers from C (in order) and C_{I^{c}} be the other numbers from C, also in order. Define the skew van der Waerden syzygy

W (T) := ∑_{I∈\binom{[p+b−a+1]}{i}}(A, C_{I^{c}})^{(a)} · (C_{I}, B)^{(b)} . (13)

Proposition 19. The syzygies W (T) form a Gröbner basis for the kernel of χ.

Proof. Our choice of term order implies in_{≺}(W (T)) = T = α^{(a)}β^{(b)}. Therefore it suffices to show that χ(W (T)) = 0. Let Y_{1}, …, Y_{m+p} be the columns of the submatrix of L given by its rows a + 1, …, b + p. The skew van der Waerden syzygy χ(W (T)) is an anti-symmetric, multilinear form in the p + b − a + 1 vectors Y_{b_{1}}, …, Y_{b_{p+b−a+1}} in (p + b − a)-space.

The non-reduced Gröbner basis in Proposition 19 can be lifted to the quantum Grassmannian as follows. We define the quantum van der Waerden syzygy of the non-standard tableau T to be the unique quadratic polynomial V (T) in kernel(φ) which satisfies

in_{w}(V (T)) = W (T),

and is a sum of syzygies S(γ^{(c)}, δ^{(d)}) with w(γ^{(c)}δ^{(d)}) = w(T). This syzygy exists by (12) and it is unique because the quadratic generators of the initial ideal are klinearly independent, and any two such quadratic lifts of W (T) in kernel(φ) differ by terms whose weights are strictly less than w(T). For instance, the quantum van der Waerden syzygy V (156^{(1)}234^{(2)}) is the polynomial with 30 terms given in (4). It would be desirable to find an explicit formula, perhaps in terms of the combinatorial formalism in [6], for all of the skew van der Waerden syzygies V (T), but at present we have no clue how to do this.

The ideal of the quantum Grassmannian K_{p,m}^{q} contains certain obvious relations which are derived from the Grassmannian Grass_{p}k^{m+p}. For each α ∈ \binom{[m+p]}{p} consider the polynomial

g_{α}(t) = α^{(q)} · t^{q} + ⋯ + α^{(1)} · t + α^{(0)} .

Given any quadratic form F (α) in the Plücker ideal defining Grass_{p}k^{m+p} and any 0 ≤ r ≤ 2q, let F_{r} be the coefficient of t^{r} in polynomial F (g_{α}(t)). Since F (g_{α}(t)) is a polynomial in t which vanishes identically on K_{p,m}^{q}, each of its coefficients F_{r} must also vanish on K_{p,m}^{q}. We call the collection of quadratic polynomials F_{r} as F ranges over a generating set for the Plücker ideal of Grass_{p}k^{m+p} the obvious relations. Rosenthal [16] showed the following.

Proposition 20. The obvious relations define K_{p,m}^{q} set-theoretically, provided k is infinite.

When p, m ≤ 2, the obvious relations coincide with the reduced Gröbner basis of Theorem 2, in particular, they generate the ideal of the quantum Grassmannian K_{2,m}^{q}. This is no longer true for m = p = 3. There are 35 incomparable pairs in C_{3,3}^{0}, and hence 35 linearly independent quadrics in the Plücker ideal of Grass_{3}k^{6}. These give rise to 35(2q + 1) linearly independent obvious relations but when q > 0 there are 35(2q + 1) + 2q − 1 incomparable pairs in C_{3,3}^{q}. Thus the obvious relations do not generate the homogeneous ideal of K_{3,3}^{q}.

When q = 1 or q = 2 then the obvious relations generate the homogeneous ideal of K_{3,3}^{q} together with an embedded component supported on the irrelevant ideal. Thus the obvious relations define K_{3,3}^{q} scheme-theoretically, but not ideal-theoretically. It remains an open problem whether the the obvious relations define K_{p,m}^{q} schemetheoretically.

Batyrev et.al. [1] applied the familiar sagbi property for the Grassmannian in the construction of certain pairs of mirror 3-folds from Calabi-Yau complete intersections in Grassmannians. We are optimistic that the results in this paper will be similarly useful for researchers in the fascinating interplay of algebraic geometry and theoretical physics.

The classical straightening law for the Grassmannian and its Schubert varieties were the starting point for the general standard monomial theory for flag varieties. For details and references we refer to the recent work on sagbi bases by Gonciulea and Lakshmibai [13]. Our results suggest that standard monomial theory might be extended to certain spaces of rational curves in flag varieties generalizing the quantum Grassmannian K_{p,m}^{q}.

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Frank Sottile, Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706, USA

  E-mail address: sottile@math.wisc.edu

Bernd Sturmfels, Department of Mathematics, University of California, Berkeley, California 94720, USA

  E-mail address: bernd@math.berkeley.edu