1 D11 D12

2

Four New Formulas for Schubert Polynomials

Peter Magyar Northeastern University pmagyar@lynx.neu.edu

June 17, 1999

We state several new combinatorial formulas for the Schubert polynomials. They are generalizations of well-known expressions for the Schur polynomials: (1) the Demazure character formula; (2) the realization as the generating function of semi-standard tableaux of a given shape; and (3), (4) the Weyl character formula. Our formulas appear surprising from a combinatorial point of view because their derivation and proof involve a new geometric model, the configuration varieties.

The results we state here are a special case of formulas for a broad class of Schur-type polynomials, the (flagged) Schur polynomials of strictly separated diagrams = [?], [?]. These include skew Schur and key polynomials [?], and the Schur polynomials of northwest diagrams [?], [?], [?], [?].

1 Schubert polynomials

The Schubert polynomials S(w) of permutations w ∈ Σ_{n} are polynomials in variables x_{1}, …, x_{n}. They were originally considered as representatives of Schubert classes in the Borel picture of the cohomology of the flag variety GL(n)/B, though we will give a completely different geometric interpretation in the later sections of this note.

They are constructed in terms of the following divided difference operators [?], [?], [?]. First, the operator ∂_{i} is defined by

∂_{i}f (x_{1}, …, x_{n}) = 3D \frac{f(x_{1}, …, x_{i}, x_{i+1}, …, x_{n})− = f(x_{1}, …, x_{i+1}, x_{i}, …, x_{n})}{x_{i} − x_{i+1}}.

Then for a reduced decomposition of a permutation u = 3Ds_{i_{1}}s_{i_{2}} ⋯, the operator ∂_{u} = 3D∂_{i_{1}}∂_{i_{2}} ⋯ is independent of the reduced decomposition chosen. Also, take ∂_{e} = 3Did.

Now we may define the Schubert polynomials as follows. Let w_{0} = 3Dn, n − 1, …, 2, 1 be the longest permutation, and take u = 3Dw^{−1}w_{0}, so that wu = 3Dw_{0}. Then

S(w) = 3D∂_{u}(x_{1}^{n−1}x_{2}^{n−2} ⋯ x_{n−2}^{2}x_{n−1}).

We have deg S(w) = 3Dℓ(w).

Example. For the permutation w = 3D24153 ∈ Σ_{5}, by inverting first ascents we get ws_{1}s_{3}s_{2}s_{1}s_{4}s_{3} = 3Dw_{0}, so

S(w) = 3D∂_{1}∂_{3}∂_{2}∂_{1}∂_{4}∂_{3}(x_{1}^{4}x_{2}^{3}x_{3}^{2}x_{4}) = 3Dx_{1}x_{2} (x_{1}x_{2}+x_{1}x_{3}+x_{2}x_{3}+x_{1}x_{4}+x_{2}x_{4}).

2 Orthodontia on a Rothe diagram

Consider the Rothe diagram of a permutation w ∈ Σ_{n},

D = 3DD(w) = 3D{(i, j) ∈ N × N | i < w^{−1}(j), j < w(i)}.

Its elements are called squares. We shall often think of D as a sequence (C_{1}, C_{2}, …, C_{r}) of columns C_{j} ⊂ N, by projecting squares (i, j) to their first coordinate. We omit any empty columns from the sequence.

In the sequel, our main interest is in Rothe diagrams, but our analysis will involve more general diagrams D ⊂ N × N of squares in the plane. In fact, to any such diagram one can associate a “Schur polynomial”, which is the character of its flagged Schur module (see Section 7). The formulas we state will apply not only to the Schubert polynomials (associated to Rothe diagrams), but to the Schur polynomials of any “strongly separated” diagram. (See Section 9.)

Now, let D be a Rothe diagram. For our formulas, we will require a sequence of permutations w_{1}, w_{2}, …, w_{r} which is compatible with D in the sense that w_{j} {1, 2, …, | C_{j}|} = 3DC_{j} for all j. We also demand that the sequence be monotone in the weak order: that is, for some u_{1}, u_{2}, …, u_{r}, we must have w_{1} = 3Du_{1}, w_{2} = 3Du_{1}u_{2}, …, w_{r} = 3Du_{1} ⋯ u_{r}, with ℓ(w_{j}) = 3Dℓ(u_{1}) + ℓ(u_{2}) + ⋯ + ℓ(u_{j}). (It is enough to require w_{r} = 3Dℓ(u_{1}) + ⋯ + ℓ(u_{r}).)

This can be done by means of the following algorithm. Given a column C ⊂ {1, …, n}, a missing tooth of C is an integer i such that i ∉ C, but i′ ∈ C for some i′ > i. The only C without any missing teeth are {1, 2, 3, …, j}. Given a diagram D = 3D(C_{1}, …, C_{r}), let (i_{0}, j_{0}) denote a special missing tooth in D which is in the leftmost column possible, and as high as possible in this column subject to the condition that (i_{0} + 1, j_{0}) ∈ D.

Now perform orthodontia on D to get a new diagram D′ with fewer missing teeth, by switching rows i_{0} and i_{0} + 1 in the columns weakly right of (i_{0}, j_{0}). That is, change D to

\begin{aligned}D′ & = 3D & \{(i, j) | (i, j) ∈ D and j < j_{0}\} \\ & & ∪\{(s_{i_{0}}i, j) | (i, j) ∈ D and j ≥ j_{0}\}. \end{aligned}

Next, locate the special missing tooth (i_{1}, j_{1}) of D′, and perform this procedure again on D′ to get D′′ and (i_{2}, j_{2}), and so on until we reach a diagram with no missing teeth. Notice that j_{0} ≤ j_{1} ≤ · · ·.

Finally, define the orthodontic sequence of D = 3DD(w) to be w_{1}, w_{2}, …, where w_{j} = 3D ∏_{k: j_{k}≤j} s_{i_{k}}, the product being taken over all k such that (i_{k}, j_{k}) is weakly left of column j. It is easily seen that this sequence has the desired properties.

Example. For the same w = 3D24153, we have

D = 3DD(w) = 3D \begin{matrix}1 & □ \\ 2 & □ & □ \\ 3 \\ 4 & & □ \end{matrix} = 3D \begin{aligned}□ & ∘ \\ □ & □ \\ & □ \end{aligned}

D′ = 3D \begin{matrix}1 & □ & □ \\ 2 & □ \\ 3 & & ∘ \\ 4 & & □ \end{matrix} D′′ = 3D \begin{matrix}1 & □ & □ \\ 2 & □ & ∘ \\ 3 & & □ \end{matrix} D′′′ = 3D \begin{matrix}1 & □ & □ \\ 2 & □ & □ \end{matrix}

so that the special missing teeth (as indicated by ∘) are (i_{0}, j_{0}) = 3D(1, 2), (i_{1}, j_{1}) = 3D(3, 2), (i_{2}, j_{2}) = 3D(2, 2), and w_{1} = 3De, w_{2} = 3Ds_{i_{0}}s_{i_{1}}s_{i_{2}} = 3Ds_{1}s_{3}s_{2}.

Note that w_{r} = 3Dw_{2} = 3Ds_{1}s_{3}s_{2} is a reduced subword of the first-ascent sequence s_{1}s_{3}s_{2}s_{1}s_{4}s_{3} which raises w to the maximal permutation w_{0}, as in the previous section. This is always the case, and we can give an algorithm for extracting this subword.

3 Demazure character formula

The definition of S(w) involves descending induction (lowering the degree), but we give the following ascending algorithm.

Let D(w) = 3D(C_{1}, …, C_{r}) (omitting empty columns), and let c_{j} = 3D| C_{j}|. Take a monotone compatible sequence w_{1}, …, w_{r} for D(w), such as the orthodontic sequence defined above, and let u_{j} = 3Dw_{j−1}^{−1}w_{j}, so that w_{1} = 3Du_{1}, w_{2} = 3Du_{1}u_{2}, · · ·. Furthermore, let λ_{i} = 3Dx_{1}x_{2} ⋯ x_{i}.

Define the Demazure operators (isobaric divided differences) π_{i} = 3D∂_{i}x_{i} and π_{u} = 3Dπ_{i_{1}}π_{i_{2}} ⋯, for u = 3Ds_{i_{1}}s_{i_{2}} ⋯ a reduced decomposition. (See [?].) These are analogous to the ∂ operators, but do not change the degree of a homogeneous polynomial.

Finally, let S_{0}(w) = 3D1 and

S_{k}(w) = 3Dπ_{u_{k}}(λ_{c_{k}}S_{k−1}(w)).

Theorem 1

S_{r}(w) = 3DS(w).

Example. For our permutation w = 3D24153, we have c_{1} = 3Dc_{2} = 3D2, u_{1} = 3De, u_{2} = 3Ds_{1}s_{3}s_{2}, and we may verify that

S(w) = 3Dx_{1}x_{2} π_{1}π_{3}π_{2}(x_{1}x_{2}).

Note that this makes the factorization evident.

4 Young tableaux

The work of Lascoux-Schutzenberger [?] and Littlemann [?] allows us to “quantize” our Demazure formula, realizing the terms of the polynomial by certain tableaux endowed with a crystal graph structure. Reiner and Shimozono have shown that our construction gives the same non-commutative Schubert polynomials as those in [?]. In fact, A. Lascoux has informed me that some of the contents of this section were known to him, and motivated [?], though not explicitly stated there. Our tableaux are different, however, from the “balanced tableaux” of Fomin, Greene, Reiner, and Shimozono.

Recall that a column-strict filling of a diagram D (with entries in {1, …, n}) is a map t filling the squares of D with numbers from 1 to n, strictly increasing down each column. The content of a filling is a monomial x^{t} = 3D ∏_{(i,j)∈D} x_{t(i,j)}, so that the exponent of x_{i} is the number of times i appears in the filling. We will define a set of fillings T of the Rothe diagram D(w) which satisfy

S(w) = 3D ∑_{t∈T} x^{t}.

The set T will be defined recursively, T_{0}, T_{1}, …, T_{r} = 3DT , so that

S_{k}(w) = 3D ∑_{t∈T_{k}} x^{t}.

We will need the root operators first defined in [?]. These are operators f_{i} which take a filling t of a diagram D either to another filling of D or to the empty set ∅. To define them we first encode a filling t in terms of its reading word: that is, the sequence of its entries starting at the upper left corner, and reading down the columns one after another: t(1, 1), t(2, 1), t(3, 1), …, t(1, 2), t(2, 2), . . ..

The lowering operator f_{i} either takes a word t to the empty word ∅, or it changes one of the i entries to i + 1, according to the following rule. First, we ignore all the entries in t except those containing i or i + 1; if an i is followed by an i + 1 (ignoring non i or i + 1 entries in between), then henceforth we ignore that pair of entries; we look again for an i followed (up to ignored entries) by an i + 1, and henceforth ignore this pair; and iterate until we obtain a subword of the form i + 1, i + 1, …, i + 1, i, i, …, i. If there are no i entries in this word, then f_{i}(t) = 3D∅, the empty word. If there are some i entries, then the leftmost is changed to i + 1.

For example, we apply f_{2} to the word

t = 3D 1 2 2 1 3 2 1 4 2 2 3 3
. 2 2 . 3 2 . . 2 2 3 3
. 2 … 2 . . 2 . . 3
. 2 … 2 . . . . . .
f_{2}(t) = 3D 1 3 2 1 3 2 1 4 2 2 3 3 =
f_{2}^{2}(t) = 3D 1 3 2 1 3 3 1 4 2 2 3 3 =
f_{2}^{3}(t) = 3D ∅

Decoding the image word back into a filling of the same diagram D, we have defined our operators.

Moreover, consider the column ϕ_{m} = 3D{1, 2, …, m} and its minimal columnstrict filling t_{m} (ith level filled with i). For a filling t of any diagram D = 3D(C_{1}, …, C_{r}), define in the obvious way the composite filling t_{m} ⊔ t of the juxtaposed diagram ϕ_{i} ⊔ D = 3D(ϕ_{i}, C_{1}, …, C_{r}). In terms of words, this means concatenating the words (1, 2, …, m) and t.

Now we can define our sets of tableau. Let our notation be as in the Demazure character formula, D(w) = 3D(C_{1}, …, C_{r}), etc, and take a reduced decomposition u_{k} = 3Ds_{i_{1}} ⋯ s_{i_{l}}. Define T_{0} = 3D{∅}, and

T_{k} = 3D⟨f_{i_{1}}⟩ ⋯ ⟨f_{i_{l}}⟩(t_{c_{k}} ⊔ T_{k−1}),

where ⟨f_{i}⟩ means the set of powers {id, f_{i}, f_{i}^{2}, …}.

Theorem 2 The Schubert polynomial S(w) is the generating function for the tableaux T = 3DT_{r}:

S(w) = 3D ∑_{t∈T} x^{t}.

Furthermore, the crystal graph structure of T reflects the splitting of S(w) into key polynomials:

S(w) = 3D ∑_{t∈Yam(T )} κ_{w_{t}(x^{t})},

where Yam(T ) is the set of Yamanouchi words in T , and the w_{t} are some permutations.

Example. As above, when c_{1} = 3Dc_{2} = 3D2, u_{1} = 3De, u_{2} = 3Ds_{1}s_{3}s_{2}, the set of tableaux (words) grows as follows:

T_{0} = 3D{∅} →^{t_{2}⊔} {12} →^{⟨f_{2}⟩} {12, 13} →^{⟨f_{3}⟩} {12, 13, 14} →^{⟨f_{1}⟩} T_{1} = 3D{12, 13, 14, 23, = 24}

→^{t_{2}⊔} T = 3DT_{2} = 3D{1212, 1213, 1214, 1223, 1224}.

This clearly gives us the Schubert polynomial as generating function, and furthermore we see the crystal graph (with vertices the tableaux in T and edges all pairs of the form (t, f_{i}t) ):

\begin{matrix}1223 & ←^{1} & 1213 \\ 3↓ & & ↓3 \\ 1224 & ←^{1} & 1214 \end{matrix} 1212

The highest-weight elements in each component are the Yamanouchi words Yam(T ) = 3D{1213, 1212}, and by looking at the corresponding lowest elements, we may deduce S(w) = 3Dκ_{x_{1}x_{2}^{2}x_{4}} + κ_{x_{1}^{2}x_{2}^{2}} = 3Dκ_{1201} + κ_{2200}. Lascoux and Schutzenberger [?] have obtained another characterization of such lowestweight tableaux.

5 Weyl character formula I

Our next character formula mixes the rational terms of the Weyl character formula with the chains of Weyl group elements in the Standard Monomial Theory of Lakshmibai-Seshadri-Musili. For computations, this formula is very inefficient: a related expression with much fewer terms, more directly generalizing the Weyl formula, is given in the following section. The main advantage of the current expression is that one can use it to obtain character formulas for certain analogous polynomials associated with other root systems (though unfortunately these analogous polynomials do not seem to include the Schubert polynomials of other root systems).

Suppose we have any permutation with choice of reduced decomposition, u = 3Ds_{i_{1}} ⋯ s_{i_{l}}, and a sequence of zeroes and ones ϵ = 3D(ϵ_{1}, ϵ_{2}, ⋯), ϵ_{j} ∈= {0, 1}. Denote

u^{ϵ} = 3Ds_{i_{1}}^{ϵ_{1}} ⋯ s_{i_{l}}^{ϵ_{l}},

a subword of u, not necessarily reduced.

Consider as before a monotone sequence of permutations w_{1}, w_{2}, …, w_{r} compatible with the Rothe diagram D = 3DD(w), and choose a reduced decomposition for u_{j} = 3Dw_{j−1}^{−1}w_{j}. This gives a choice of reduced decomposition for each w_{j} = 3Du_{1} = ⋯ u_{j}, and in particular for w_{r} = 3Ds_{i_{1}} ⋯ s_{i_{l}}. Let v_{k} = 3Ds_{i_{1}} ⋯ s_{i_{k}} for k ≤ l. Recall that λ_{c_{j}} = 3Dx_{1}x_{2} ⋯ x_{c_{j}}, the fundumental weight associated to the length c_{j} of the jth column of D (not counting empty columns).

Theorem 3

S(w) = 3D ∑_{ϵ} \frac{∏_{j=3D1}^{r} w_{j}^{ϵ}(λ_{c_{j}})}{∏_{k=3D1}^{l} (1 − v_{k}^{ϵ}(x_{i_{k}}^{−1}x_{i_{k}+1})),}

where the summation is over all 2^{l} sequences of zeroes and ones ϵ = 3D(ϵ_{1}, …, ϵ_{l}).

This follows straight-forwardly from our Demazure formula, though it also has a geometric interpretation in terms of Bott-Samelson varieties (see below). Example. For the same w = 3D24153, we have the reduced decomposition w_{r} = 3Dw_{2} = 3Ds_{1}s_{3}s_{2}, l = 3D3, so the Schubert polynomial is the following sum of 8 terms corresponding to ϵ = 3D(000), (001), (010), (011), …:

S(w) = 3D \frac{x_{1}^{2}x_{2}^{2}}{(1−x_{1}^{−1}x_{2})(1−x_{3}^{−1}x_{4})(1−x_{2}^{−1}x_{3})} + \frac{x_{1}^{2}x_{2}x_{3}}{(1−x_{1}^{−1}x_{2})(1−x_{3}^{−1}x_{4})(1−x_{3}^{−1}x_{2})}

+\frac{x_{1}^{2}x_{2}^{2}}{(1−x_{1}^{−1}x_{2})(1−x_{4}^{−1}x_{3})(1−x_{2}^{−1}x_{4})} + \frac{x_{1}^{2}x_{2}x_{4}}{(1−x_{1}^{−1}x_{2})(1−x_{4}^{−1}x_{3})(1−x_{4}^{−1}x_{2})}

+\frac{x_{1}^{2}x_{2}^{2}}{(1−x_{2}^{−1}x_{1})(1−x_{3}^{−1}x_{4})(1−x_{1}^{−1}x_{3})} + \frac{x_{1}x_{2}^{2}x_{3}}{(1−x_{2}^{−1}x_{1})(1−x_{3}^{−1}x_{4})(1−x_{3}^{−1}x_{1})}

+\frac{x_{1}^{2}x_{2}^{2}}{(1−x_{2}^{−1}x_{1})(1−x_{4}^{−1}x_{3})(1−x_{1}^{−1}x_{4})} + \frac{x_{1}x_{2}^{2}x_{4}}{(1−x_{2}^{−1}x_{1})(1−x_{4}^{−1}x_{3})(1−x_{4}^{−1}x_{1})}.

6 Weyl Character Formula II

Finally, we state a result directly generalizing the Weyl character formula (Jacobi bialternant), reducing to it in case S(w) is a Schur polynomial.

The formula involves certain extensions of the Rothe diagram D = 3DD(w). Define the Young diagram Φ = 3D{(i, j) | 1 ≤ i ≤ j ≤ n − 1}. Let the flagged diagram Φ ⊔ D be the concatenation of the two diagrams placed horizontally next to each other: that is, the columns of Φ ⊔ D are those of Φ followed by those of D.

Now, given Φ ⊔ D = 3D(C_{1}, …, C_{r}), define the blowup of the flagged diagram Φ̂ ⊔ D = 3D(C_{1}, …, C_{r}, C′_{1}, C′′_{2}, …), where the extra columns are the intersections C̃ = 3DC_{i_{1}} ∩ C_{i_{2}} ∩ ⋯ ⊂ N, for all lists C_{i_{1}}, C_{i_{2}}, … of columns of Φ ⊔ D; but if an intersection C_{i_{1}} ∩ C_{i_{2}} ∩ ⋯ = 3DC_{k} is already a column of Φ ⊔ D, then we do not append it.

Now let D̃ = 3DΦ̂ ⊔ D. Define a standard tabloid t of D̃ to be a columnstrict filling such that if C, C′ are columns of D̃ with C horizontally contained in C′, then the numbers filling C all appear in the filling of C′. In symbols, t : D̃ → {1, …, n} , t(i, j) < t(i + 1, j) for all i, j, and C ⊂ C′ ⇒ t(C) ⊂ t(C′).

For 1 ≤ i ≠ j ≤ n and a tabloid t of D̃, we define certain integers: d_{ij}(t) is the number of connected components of the following graph. The vertices are columns C of D̃ such that i ∈ t(C), j ∉ t(C); the edges are (C, C′) such that C ⊂ C′ or C′ ⊂ C.

Finally, since there are inclusions of diagrams D, Φ ⊂ D̃ = 3DΦ̂ ⊔ D, we have the restrictions of a tabloid t for D̃ to D and Φ, which we denote t|D and t|Φ.

Theorem 4

S(w) = 3D ∑_{t} \frac{x^{(t|D)}}{∏_{i<j} (1 − x_{i}^{−1}x_{j})^{d_{ij}(t)−1} (1 − x_{j}^{−1}x_{i})^{d_{ji}(t)}} ,

where t runs over the standard tabloids for Φ̂ ⊔ D such that (t|Φ)(i, j) = 3Di for all (i, j) ∈ Φ.

Example. For the same w = 3D24153,

D = 3DD(w) = 3D \begin{matrix}1 & □ \\ 2 & □ & □ \\ 3 \\ 4 & & □ \end{matrix} Φ ⊔ D = 3D \begin{aligned}□ & □ & □ & □ & □ \\ & □ & □ & □ & □ & □ \\ & & □ & □ \\ & & & □ & & □ \end{aligned}

Φ̂ ⊔ D = 3D \begin{matrix}1 & □ & □ & □ & □ & □ \\ 2 & & □ & □ & □ & □ & □ & □ \\ 3 & & & □ & □ \\ 4 & & & & □ & & □ \end{matrix} .

There are six standard tabloids of the type occurring in the theorem. Their restrictions to the last three columns of Φ̂ ⊔ D are:

1 2 1 1 2 , 1 2 1 1 3 , 1 2 1 1 4 , 1 2 1 2 2 , 1 2 2 2 3 , 1 2 2 2 4 .

The integers d_{ij}(t) are 0, 1, or 2, and we obtain

S(w) = 3D \frac{x_{1}^{2}x_{2}^{2}}{(1−x_{1}^{−1}x_{2})(1−x_{2}^{−1}x_{3})(1−x_{2}^{−1}x_{4})} + \frac{x_{1}^{2}x_{2}x_{3}}{(1−x_{1}^{−1}x_{2})(1−x_{3}^{−1}x_{4})(1−x_{3}^{−1}x_{2})}

+\frac{x_{1}^{2}x_{2}x_{4}}{(1−x_{1}^{−1}x_{2})(1−x_{4}^{−1}x_{2})(1−x_{4}^{−1}x_{3})} + \frac{x_{1}^{2}x_{2}^{2}}{(1−x_{1}^{−1}x_{3})(1−x_{1}^{−1}x_{4})(1−x_{2}^{−1}x_{1})}

+\frac{x_{1}x_{2}^{2}x_{3}}{(1−x_{2}^{−1}x_{1})(1−x_{3}^{−1}x_{4})(1−x_{3}^{−1}x_{1})} + \frac{x_{1}x_{2}^{2}x_{4}}{(1−x_{2}^{−1}x_{1})(1−x_{4}^{−1}x_{1})(1−x_{4}^{−1}x_{3})}.

Note that some, but not all, of the above six terms are among the eight terms of the previous example. As in the case of the original Weyl character formula, it is not clear a priori why either of these rational functions should simplify to a polynomial (with positive integer coefficients).

This formula has been implemented by the author in Mathematica, = available on request.

7 Schur modules

The above formulas arise naturally from a Borel-Weil theory which relates Schubert polynomials with certain algebraic varieties similar to the Schubert varieties of GL(n). (See [?].) The starting point of our theory is the result of Kraskiewicz and Pragacz which realizes Schubert polynomials as characters of “flagged Schur modules”.

We shall write G = 3DGL(n, C), B = 3D the subgroup of upper triangular matrices, V = 3DC^{n} the defining representation, and V_{i} the subspace of V spanned by the first i coordinate vectors.

Let the diagram D ⊂ N × N be any set of squares (i, j) in the plane. Let Σ_{D} be the symmetric group permuting the squares of D, Col(D) ⊂ Σ_{D} the subgroup permuting the squares within each column, and Row(D) similarly for rows. Define (almost) idempotents α_{D}, β_{D} in the group algebra C[Σ_{D}] by

α_{D} = 3D ∑_{σ∈Row D} σ, β_{D} = 3D ∑_{σ∈Col D} sgn(σ)σ,

where sgn(σ) is the sign of the permutation.

Σ_{D} acts on the right of the tensor product V^{⊗D} by permuting factors, and G and B act on the left by the diagonal action. These two actions commute. Define the flagged Schur module to be the B-stable subspace

S_{D}^{B} = 3D( ⨂_{(i,j)∈D} V_{i} ) α_{D}β_{D} ⊂ V^{⊗D}.

Theorem 5 (Kraskiewicz-Pragacz) The Schubert polynomial for a permutation is the character of the flagged Schur module of its Rothe diagram:

S(w) = 3D tr( diag(x_{1}, …, x_{n}) | S_{D(w)}^{B}).

Here the constant n is taken so that D(w) has at most n rows.

Reiner, Shimozono, and the author have given a proof of this theorem using configuration varieties.

Example. For w = 3D24153, if we number the squares of D(w) as

D = 3DD(w) = 3D \begin{aligned}□ \\ □ & □ \\ & □ \end{aligned} = 3D 1 2 3 4 ,

then we have (in cycle notation for Σ_{D} ≅ Σ_{4}),

\begin{aligned}α_{D}β_{D} & = 3D & (1 + (23)) (1 − (12))(1 − (34)) \\ & = 3D & 1 + (23) − (12) − (34) − (132) − (234) + (12)(34) + (1324). \end{aligned}

Take n = 3D5, so that V^{⊗D} ≅ (C^{5})^{⊗4}, a 20-dimensional space with coordinate vectors e_{i_{1}i_{2}i_{3}i_{4}} = 3De_{i_{1}} ⊗ e_{i_{2}} ⊗ e_{i_{3}} ⊗ e_{i_{4}}, i_{1}, …, i_{4} ∈ {1, …, 5}. Σ_{D} acts by e_{i_{1}i_{2}i_{3}i_{4}} · σ = 3De_{i_{σ(1)}i_{σ(2)}i_{σ(3)}i_{σ(4)}}, and GL(5) acts diagonally on the tensor factors. By definition, S_{D}^{B} is spanned by vectors of the form

v_{t} = 3De_{t(1),t(2),t(3),t(4)} α_{D}β_{D},

for all fillings t : D → {1, …, 5} with t(1) ≤ 1, t(2), t(3) ≤ 2, t(4) ≤ 4, and it should follow from our theory that we obtain a basis if we take only the 6 fillings t ∈ T , the set of tableaux defined in our second character formula. For instance, if t = 3D(1213),

\begin{aligned}v_{t} & = 3D & e_{1213} + e_{1123} − e_{2113} − e_{1231} − e_{1123} − e_{1132} + e_{2131} + e_{1132} \\ & = 3D & e_{1213} − e_{2113} − e_{1231} + e_{2131}. \end{aligned}

8 Configuration varieties

Now we translate our algebra into geometry, realizing a flagged Schur module S_{D}^{B} as the space of sections of a line bundle over a (possibly singular) algebraic variety F_{D}^{B}. Because the singularities are sufficiently tame, we can obtain non-trivial transformations of our problem by considering desingularizations of F_{D}^{B} and applying known character formulas for spaces of sections over smooth varieties. The first three formulas come from a Bott-Samelson resolution of F_{D}^{B} [?], the last from an analog of Zelevinsky’s resolution [?].

Let D ⊂ N × N be a diagram with all its squares in rows i = 3D1, …, n. Let C_{1}, …, C_{r} ⊂ {1, …, n} be the columns of D, and for each C = 3DC_{j}, let V_{C} = 3DSpan {e_{i} | i ∈ C} ⊂ C^{n}, a coordinate subspace of dimension c_{j} = 3D|C_{j}|. Consider the = r-tuple (V_{C_{1}}, …, V_{C_{r}}) as a point in the product of Grassmannians Gr(D) = 3D Gr(c_{1}, C^{n}) × ⋯ × Gr(c_{r}, C^{n}), and define the flagged configuration variety

F_{D}^{B} = 3D\overline{B · (V_{C_{1}}, …, V_{C_{r}})} ⊂ Gr(D),

the closure of the B-orbit of the above point. This is an irreducible projective variety, and the Schubert varieties of GL(n)/B and GL(n)/P are clearly special cases. F_{D}^{B} has a natural line bundle L_{D} defined by restricting the Plucker bundle O(1, …, 1) over the product of Grassmannians Gr(D). These varieties are very tractable in the case of a Rothe diagram, and we may state the following Borel-Weil-Bott theorem.

Theorem 6 (Magyar-van der Kallen) Let D = 3DD(w) a Rothe diagram. Then F_{D}^{B} has rational singularities and is projectively normal with respect to L_{D}. Furthermore, the space of global sections

H^{0}(F_{D}^{B}, L_{D}) ≅ (S_{D}^{B})^{∗}

as B-modules, and H^{i}(F_{D}^{B}, L_{D}) = 3D0 for all i > 0.

Now let w_{1}, …, w_{r} be the orthodontic sequence of D = 3DD(w) and w_{r} = 3Ds_{i_{1}} ⋯ s_{i_{l}} the associated reduced decomposition. Thus the initial subwords are reduced decompositions w_{j} = 3Ds_{i_{1}} ⋯ s_{i_{l(j)}}, where l(j) = 3Dl(w_{j}). Consider the associated Bott-Samelson variety [?]

Z = 3DP_{i_{1}} ×^{B} ⋯ ×^{B} P_{i_{l}}/B,

where P_{i} denotes the maximal parabolic of G = 3DGL(n) such that G/P_{i} ≅ Gr(i, C^{n}). Define the multiplication map

\begin{aligned}ϕ : & Z & → & Gr(D) ≅ G/P_{c_{1}} × G/P_{c_{2}} × ⋯ × G/P_{c_{r}} = \\ & (p_{1}, …, p_{l}) & ↦ & (p_{1}p_{2} ⋯ p_{l(1)}, p_{1}p_{2} ⋯ p_{l(2)}, …, p_{1}, ⋯ p_{l(r)}). \end{aligned}

Theorem 7 The map ϕ maps Z birationally onto F_{D}^{B}, and so is a resolution of singularities. Furthermore, for all i,

H^{i}(Z, ϕ^{∗}L_{D}) ≅ H^{i}(F_{D}^{B}, L_{D}).

From this, the computations of Demazure [?] on the Bott-Samelson variety directly imply our formula (1), and (2) follows by the theory of root operators. Formula (3) results from applying the Atiyah-Bott-Lefschetz fixed-point formula to Z.

Theorem 8 Let D = 3DD(w) and D̃ = 3DΦ̂ ⊔ D the blowup diagram of section ??. Then the configuration variety F B ; _{D} is a smooth variety, and is a reso-

lution of singularities of F_{D}^{B} via the natural projection map map ψ : Gr(D̃) → Gr(D). Furthermore, for all i,

H^{i}(F B ; _{D}, ψ^{∗}L_{D}) ≅ H^{i}(F_{D}^{B}, L_{D}).

Formula (4) now results from Atiyah-Bott-Lefschetz applied to F B ; _{D}.

Example. In our case w = 3D24153, we may take n = 3D4, so that we have Gr(D) = 3D Gr(2, C^{4}) × Gr(2, C^{4}), which we may think of as the variety of pairs of lines in P^{3}. The B-orbit of the special point (V_{12}, V_{24}) is precisely the pairs of the form (V_{12}, W), where W = 3D⟨v_{1}, v_{2} =⟩, v_{1} ∈ V_{12}, and v_{1}, v_{2} are linearly independent. Thus

F_{D}^{B} ≅ {W ∈ Gr(2, C^{4}) | dim(V_{12} ∩ W) ≥ 1},

the variety of lines in P^{3} which intersect the coordinate axis. (We can give such a description of F_{D}^{B} as configurations with intersection conditions for any Rothe diagram. ) This is the singular Schubert variety in Gr(2, C^{4}), the resolutions mentioned are the original Bott-Samelson and Zelevinsky resolutions, and there exist regular maps Z → F B ; _{D} → F_{D}^{B}.

P^{3}plane}.

9 Generalizations

The above results can be used to compute the characters of Schur modules more general than those associated to Rothe diagrams. In fact, let us replace D(w) by any diagram which satisfies the following strictly separated condition. For two sets S, S′ ⊂ N, we say S < S′ if s < s′ for all s ∈ S, s′ ∈ S′. Now, a diagram D = 3D(C_{1}, …, C_{r}) with columns C_{j} ⊂ {1, …, n}, is strictly separtated if, for any two columns C, C′ of D, we have

(C \ C′) < (C′ \ C) or (C \ C′) > (C′ \ C) ,

where C \ C′ denotes the complement of C′ in C. See [?], [?]. Replace the Schubert polynomial by the character of the flagged Schur module S_{D}^{B}. Then our Theorems 1, 2, 3, 6, and 7 remain valid: the orthodontic algorithm gives a sequence of Weyl group elements compatible with D, the first three formulas compute the character of S_{D}^{B}, and the associated configuration variety F_{D}^{B} satisfies the Borel-Weil Theorem and possesses a Bott-Samelson resolution.

Suppose that D satisfies an even stronger property, the northwest condition:

(i, j), (i′, j′) ∈ D ⇒ ( min(i, i′), min(j, j′) ) ∈ D = .

Then Theorems 4 and 8 are valid as well: the fourth character formula is true for S_{D}^{B}, and the variety has a Zelevinsky resolution smaller than the Bott-Samelson one.

References

[1] A. Berenstein, S. Fomin, and A. Zelevinsky, Parametrizations of canonical bases and totally positive matrices, preprint 1995.

[2] M. Demazure, Désingularisation des variétés de Schubert généralisés, Ann. Sci. Ec. Norm. Sup 7 (1974), 53-88.

[3] M. Demazure, Une nouvelle formule des caractères, Bull. Sci. Math. (2) 98 (1974), 163-172.

[4] W. Kraskiewicz and P. Pragacz, Foncteurs de Schubert, C.R. Acad. Sci. Paris 304 Ser I No 9 (1987), 207-211.

[5] A. Lascoux and M.-P. Schutzenberger, Polynomes de Schubert, C. R. Acad. Sci. Paris 294 (1982), 447-450.

[6] A. Lascoux and M.-P. Schutzenberger, Tableaux and non-commutative Schubert polynomials, Funkt. Anal. 23 (1989), 63-64.

[7] A. Lascoux and M.-P. Schutzenberger, Keys and Standard Bases, Tableaux and Invariant Theory, IMA Vol. in Math. and App. 19, ed. D. Stanton (1990), 125-144.

[8] B. LeClerc and A. Zelevinsky, Chamber sets, preprint.

[9] P. Littelmann, A Littlewood-Richardson rule for Symmetrizable Kac-Moody algebras, Inv. Math. 116 (1994), 329-346.

[10] I.G. Macdonald, Notes on Schubert Polynomials, Pub. LCIM 6, Univ. du Québec a Montréal, 1991.

[11] P. Magyar, Borel-Weil theorem for configuration varieties and Schur modules, preprint alg-geom/9411014.

[12] P. Magyar, Bott-Samelson Varieties and Configuration Spaces, preprint alggeom/9611019.

[13] V. Reiner and M. Shimozono, Specht series for column-convex diagrams, to appear in J. Algebra.

[14] V. Reiner and M. Shimozono, Key polynomials and a flagged Littlewood-Richardson rule, J. Comb. Th. Ser. A 70 (1995), 107-143.

[15] A. Zelevinsky, Small resolutions of singularities of Schubert varieties, Funct. Anal. App. 17 (1983), 142-144.