The Prism tableau model for Schubert polynomials

The Schubert polynomials lift the Schur basis of symmetric polynomials into a basis for Z[x1,x2,...]. We suggest the"prism tableau model"for these polynomials. A novel aspect of this alternative to earlier results is that it directly invokes semistandard tableaux; it does so as part of a colored tableau amalgam. In the Grassmannian case, a prism tableau with colors ignored is a semistandard Young tableau. Our arguments are developed from the Groebner geometry of matrix Schubert varieties.


Overview
A. Lascoux-M.-P.Schützenberger [LaSh82a] recursively defined an integral basis of Pol = Z[x 1 , x 2 , . ..] given by the Schubert polynomials {S w : w ∈ S ∞ }.If w 0 is the longest length permutation in the symmetric group S n then S w0 := x n−1 Otherwise, w = w 0 and there exists i such that w(i) < w(i + 1).Now one sets S w = ∂ i S wsi , where ∂ i f := f −sif xi−xi+1 (since the polynomial operators ∂ i form a representation of S n , this definition is self-consistent.)It is true that under the standard inclusion ι : S n → S n+1 , S w = S ι(w) .Thus one can refer to S w for each w ∈ S ∞ = n≥1 S n .
Textbook understanding of the ring Sym of symmetric polynomials centers around the basis of Schur polynomials and its successful companion, the theory of Young tableaux.Since Schur polynomials are instances of Schubert polynomials, the latter basis naturally lifts the Schur basis into Pol.Yet, it is also true that Schubert polynomials have nonnegative integer coefficients.Consequently, one has a natural problem: Is there a combinatorial model for Schubert polynomials that is analogous to the semistandard tableau model for Schur polynomials?Indeed, multiple solutions have been discovered over the years, e.g., [Ko90], [BiJoSt93], [BeBi93], [FoSt94], [FoKi96], [FoGrReSh97], [Ma98], [BeSo98,BeSo02], [BuKrTaYo04] and [CoTa13] (see also [LaSh85]).In turn, the solutions [BiJoSt93,BeBi93,FoSt94,FoKi96] have been the foundation for a vast literature at the confluence of combinatorics, representation theory and combinatorial algebraic geometry.
We wish to put forward another solution -a novel aspect of which is that it directly invokes semistandard tableaux.Both the statement and proof of our alternative model build upon ideas about the Gröbner geometry of matrix Schubert varieties X w .We use the Gröbner degeneration of X w and the interpretation of S w as mutidegrees of X w [KnMi05].Actually, a major purpose of loc.cit. is to establish the geometric naturality of the combinatorics of [BiJoSt93,BeBi93,FoKi96].Our point of departure is stimulated by later work of A. Knutson on Frobenius splitting [Kn09, Theorem 6 and Section 7.2].

The main result
We recall some permutation combinatorics found in, e.g., in [Ma01].The diagram of w is D(w) = {(i, j) : 1 ≤ i, j ≤ n, w(i) > j and w −1 (j) > i} ⊂ n × n.Let Ess(w) ⊂ D(w) be the essential set of w: the southeast-most boxes of each connected component of w.The rank function is r w (i, j) = #{t ≤ i : w(t) ≤ j}.
Define w to be Grassmannian if it has at most one descent, i.e., at most one index k such that w(k) > w(k + 1).If in addition w −1 is Grassmannian then w is biGrassmannian.For e = (i, j) ∈ Ess(w) let R e be the (i − r w (i, j)) × (j − r w (i, j)) rectangle with southwest corner at position (i, 1) of n × n.The shape of w is λ(w) = e∈Ess(w) R e : Fig. 1: The diagram of w = 35142 (with color coded essential set {e1, e2, e3}), the overlay of Re 1 , Re 2 , Re 3 , and the shape λ(w).
Let d i (w) be the number of distinct values (ignoring color) seen on the i-th antidiagonal (i.e., the one meeting (i, 1)), for i = 1, 2, . . ., n.We say T is minimal if .
Forgetting colors gives the following expansion of the Schur polynomial: In general, if w is Grassmannian then λ(w) is a (French) Young diagram.Moreover, each cell of T ∈ Prism(w) uses only one number.Replacing each set in T by the common value gives a reverse semistandard tableau.Thus P w = s λ (w) follows.
Prism tableaux provide a means to understand the RC-graphs of [BeBi93,FoKi96].We think of the #Ess(w)-many semistandard tableaux of a prism tableau T as the "dispersion" of the associated RCgraph through T .See Sections 4.1 and 4.3.
Minimality and the unstable triple condition bond the tableau of each color, which is one reason why we prefer not to think of a prism tableau as merely a #Ess(w)-tuple: Example 1.3 (Unstable triples).Let w = 42513.Then #Ess(w) = 3.The minimal prism tableaux and their weights are: The second and the fourth tableaux have an unstably paired label.In both tableaux, the pink 1 in the second antidiagonal is replaceable by a pink 2.

Main idea of the model and its proof
Let G = GL n and B and B + the Borel subgroups of lower and upper triangular matrices in G. Identify the flag variety with the coset space B\G.Let T be the maximal torus in B. Suppose X ⊂ B\G is an arbitrary subvariety and π : G B\G is the natural projection.Then carries a left B action and thus the action of T. Therefore, one can speak of the equivariant cohomology class Moreover, the polynomial [X] T is a coset representative under the Borel presentation of where I Sn is the ideal generated by (non-constant) elementary symmetric polynomials (cf.[KnMi05, pg. 1280]).This is a key perspective of work of A. Knutson-E.Miller [KnMi05] when X is a Schubert variety.Let Y ⊆ Mat n×n be an equidimensional, reduced union of coordinate subspaces.Given P ⊂ n × n, we represent P visually as a collection of +'s in the n × n grid.We say P is a plus diagram for Y , if Let Plus(Y ) be the set of all such plus diagrams.Let MinPlus(Y ) be the set of minimal plus diagrams, i.e., those P for which removing any + would not return an element of Plus(Y ).We refer to the union of plus diagrams as an overlay to emphasize whenever (i, j) is in P or P , the diagram for P ∪ P also has a + in position (i, j).
Each P corresponds 1 : 1 to a face of the Stanley-Reisner complex ∆ Y .Let ∆ n×n be the power set of {(i, j) : 1 ≤ i, j ≤ n}.Then ∆ Y ⊆ ∆ n×n and for each P one has the face The faces of ∆ Y are ordered by reverse containment of their plus diagrams.Thus, facets (maximal dimensional faces) of ∆ Y coincide with elements of MinPlus(Y ).In addition, taking the overlay of P ∈ Plus(Y ) and Q ∈ Plus(Z) corresponds to intersecting faces in the Stanley-Reisner complex:

Through the interpretation of [Y ]
T as a multidegree, one may express [Y ] T as a generating series over MinPlus(Y ).That is, (2.1) For details, the reader may consult [MiSt05]; see Chapter 1 and Chapter 8 (and its notes).Suppose ≺ is any term order on C[Mat n×n ] and X Suppose X is reduced, and hence a reduced union of coordinate subspaces.Since X was assumed to be irreducible, then X is irreducible.So by [KaSt95, Theorem 1] the Stanley-Reisner complex ∆ X of X is equidimensional.Hence we may apply the discussion above using We are interested in understanding ∆ X under certain hypotheses on X. Assume that we have a collection of varieties X, X 1 , . . ., Assume ≺ is a term order on C[V ] that defines a Gröbner degeneration of these varieties so that each Gröbner limit (2.4) Call {X i } a ≺-spectrum for X.To construct a cheap example, pick any Gröbner basis G = {g 1 , . . ., g M } with square-free lead terms to define X.
and set X k to be cut out by G k .On the other hand, a motivating example is A. Knutson [Kn09, Theorem 6]: given a term order ≺ (satisfying a hypothesis), there is a stratification of V into a poset of varieties (ordered by inclusion) with the additional feature that each stratum X admits a ≺-spectrum using higher strata.
How can a ≺-spectrum be used to understand the combinatorics of [X ] T ?Here is a simple observation: Proof.(I): Let P ∈ Plus(X ).Then L P ⊆ X ⊆ X i for all i.Therefore P ∈ Plus(X i ) and trivially P = P ∪ . . .∪ P, proving "⊆".For the other containment, suppose P i ∈ Plus(X i ) for 1 ≤ i ≤ k and let P = P 1 ∪ • • • ∪ P k .Then L P = L P1 ∩ ... ∩ L P k and hence L P ⊆ L Pi ⊆ X i .So P ∈ Plus(X i ) for each i, which implies P ∈ Plus(X ).
(II): Let P ∈ MinPlus(X ).By (I), P ∈ Plus(X i ) for each i.Then there exists P i ∈ MinPlus(X i ) so that P i ⊆ P. Then P ⊇ P 1 ∪ • • • ∪ P k ∈ Plus(X ) by (I).As P is minimal, this is an equality.
Our point is that in good cases, the plus diagrams of X i are "simpler" to understand than those of X. Lemma 2.1(II) says that one can think of each P ∈ MinPlus(X) as an overlay P = P 1 ∪ • • • ∪ P k of these simpler P i .Of course, this representation is not unique in general, so one can make a choice of representation for each P. The hope is to transfer understanding of the combinatorics of MinPlus(X i ) to the combinatorics of MinPlus(X).

Proof of the Theorem 1.1
We now carry out the ideas described in Section 2 in the case of Schubert varieties.

Matrix Schubert varieties and Schubert polynomials
The flag variety B\G decomposes into Schubert cells X • w := B\BwB + indexed by w ∈ S n .The Schubert variety is the Zariski-closure X w := X • w .The matrix Schubert variety is Let Z = (z ij ) 1≤i,j≤n be the generic n × n matrix.The Schubert determinantal ideal is In [Fu91, Lemma 3.10] it is proved that I w cuts out X w scheme-theoretically. Moreover in loc.cit. it is shown that I w is generated by the smaller set of generators coming from those (i, j) ∈ Ess(w).By [KnMi05, Theorem A], Moreover, let ≺ anti be any antidiagonal term order on C[Mat n×n ], i.e., one that picks off the antidiagonal term of any minor of Z.In [KnMi05, Theorem B] it is shown that MinPlus(X w ) are in a transparent bijection with the RC-graphs of [BeBi93] (cf.[FoKi96]).
Call {X u1 , . . ., X u k } the biGrassmannian ≺ anti -spectrum for X w .By [Kn09, Section 7.2], {X ui } indeed gives a ≺ anti -spectrum for X w over Q.This result can also be readily obtained (over Z) if one assumes the Gröbner basis result [KnMi05,Theorem B].(It should be emphasized that one of the points of [Kn09, Section 7.2] is to reprove said Gröbner basis theorem more easily.)

Multi-plus diagrams
The technical core of our proof is to analyze the combinatorics of overlays of plus diagrams for the biGrassmannian ≺ anti -spectrum {X u1 , . . ., X u k }.Let MinPlus(X ui ) be the set of multi-plus diagrams for w: we represent (P 1 , . . ., P k ) ∈ Multi(w) as a placement of colored +'s in a single n × n grid, where (a, b) has a + of color u i if (a, b) ∈ P i .

• • • • • • •
Fig. 2: The Stanley-Reisner complexes for X 1423 and X 2314 intersect to give the complex for X 2413 .These complexes are a multicone over the depicted complex.

Local moves on plus diagrams
A southwest move is the following local operation on a plus diagram: The first rule for Schubert polynomials was conjectured by [Ko90].This rule begins with the diagram of w and evolves other subsets of n × n by a simple move, the Schubert polynomial is a generating series over these subsets.A proof is presented in [Wi99,Wi02].Arguably, this rule is the most handy of all known rules, even though the set of Kohnert diagrams does not have a closed description.
Probably the most well-known and utilized formula is given by [BiJoSt93], which expresses the Schubert polynomial in terms of reduced decompositions of w.This rule is made graphical by the RC-graphs of [BeBi93] (cf.[FoKi96]).One can obtain any RC-graph for w from any other by the chute and ladder moves of [BeBi93].
While neither of the above rules transparently reduces to the tableau rule for Schur polynomials, it is not too difficult to show in either case, that the objects involved do biject with semistandard tableaux, see [Ko90] and [Ko00] respectively.
We are not aware of any published bijection between the Kohnert rule and any other model for Schubert polynomials.On the other hand, there is a map between the prism tableaux and RC-graphs: the labels on the i-th antidiagonal indicate the row position of the +'s on the same antidiagonal in the associated RC-graph.We treat each RC-graph as a specific overlay of RC-graphs for bigrassmannian permutations.The latter RC-graphs are in bijection with semistandard tableaux of rectangular shape.This is the reason for the "dispersion" remark of the introduction.
The work of [FoGrReSh97] gives a tableau rule for Schubert polynomials of a different flavor.This rule treats S w as a generating series for balanced fillings of the diagram of w.The reduction to semistandard tableaux for Grassmannian w seems non-trivial.
In [BuKrTaYo04], a formula is given for a Schubert polynomial as a nonnegative integer linear combination of sum of products of Schur functions in disjoint sets of variables (with nontrivial coefficients).This is also in some sense a tableau formula for S w .In [Le04] this result is rederived as a consequence of the crystal graph structure on RC-graphs developed there.

Stable Schubert polynomials
The stable Schubert polynomial (also known as the Stanley symmetric polynomial) is the generating series defined by F w (x 1 , x 2 , . ..) := lim m→∞ S 1 m ×w , where if w ∈ S n then 1 m × w is the permutation in S m+n defined by It is true that F w (x 1 , x 2 , . . ., x m , 0, 0, . ..) = S 1 m ×w (x 1 , . . ., x m , 0, 0, . ..).Now, notice that λ(1 m × w) and λ(w) are the same shape, but the former is shifted down m steps in the grid relative to λ(w).Therefore it follows that where the sum is over all unflagged (i.e., exclude (S4)) minimal prism tableaux of shape λ(w) that use the labels 1, 2, . . ., m.In the limit, this argument implies the generating series F w (x 1 , x 2 , . ..) is given by the same formula, except we allow all labels from N.

An overlay interpretation of chute and ladder moves
In [BeBi93], chute moves were defined for pipe dreams.These moves are locally of the form Suppose P ∈ MinPlus(X w ), biGrass(w) = {u 1 , . . ., u k } and P = P 1 ∪ • • • ∪ P k , where P i ∈ MinPlus(X ui ).The following claims show: The chute move's "long jump" of a single + may be interpreted as a sequence of the northeast local moves (3.1) applied to the P i 's.
Let the support of the first and third plus diagrams be P and Q, respectively.We have P, Q ∈ MinPlus(X w ).P and Q differ by a chute move.At the level of the overlays, one sees this transition as an application of (3.1) to each blue + in the second row.
Example 4.1 indicates the general pattern.Let (i, j) be the position of the southwest + of P in (4.1) and (i − 1, j ) the position of the northeast + of Q in (4.1).Without loss of generality, we may assume each P 1 , . . ., P t contains a + at (i, j) while all other P h do not.
Claim 4.2.Consider the interval of consecutive +'s in row i of P h (1 ≤ h ≤ t) starting at the left with the + in position (i, j).One can apply the move (3.2) (in the right to left order) to each of these +'s to obtain P 1 , . . ., P t .Claim 4.3.P h ⊆ Q for 1 ≤ h ≤ t and P h ⊆ Q for t + 1 ≤ h ≤ k.A similar discussion applies to the ladder moves.

Future work
It is straightforward to assign weights to prism tableau in order to give a formula for double Schubert polynomials.
A generalization to Grothendieck polynomials requires a deeper control of the overlay procedure.In investigating this, one is led to some results of possibly independent interest.Specifically, for Theorem 1.1, we have used the fact that the facets of ∆ X w are intersections of facets of those associated to biGrass(w).One can make a similar conjecture for all interior faces w's complex.Each ∆ X w is a ball or sphere [KnMi04,Theorem 3.7].Hence one can refer to the interior faces of this complex.Let IntPlus(w) = {P : P ∈ Plus(w) and F P is an interior face of ∆ X w }.
Conjecture 4.5.IntPlus(w) ⊆ {P 1 ∪ • • • ∪ P k : P i ∈ IntPlus(u i ), for u i ∈ biGrass(w)}.Conjecture 4.5 has been exhaustively computer checked for all n ≤ 6.As part of an intended proof of Conjecture 4.5, one defines K-theoretic analogues of the chute and ladder moves of [BeBi93]: that is if P → Q by a chute move (respectively, ladder move) then P → P ∪Q is a K-chute (respectively, K-ladder move).Whereas not all interior plus diagrams are connected by the original chute and ladder moves, it is true that they are connected once one allows the extended moves.
The first author plans to address these and related issues elsewhere.
where (w) is the Coxeter length of w.Let c be a label of color c.Labels { c , d , e } in the same antidiagonal form an unstable triple if < and replacing the c with c gives a prism tableau.See Example 1.3.