Determinantal identities for flagged Schur and Schubert polynomials
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Abstract
We prove new determinantal identities for a family of flagged Schur polynomials. As a corollary of these identities we obtain determinantal expressions of Schubert polynomials for certain vexillary permutations.
Keywords
Schur polynomial Schubert polynomial Young tableau Plane partitionMathematics Subject Classification
05E05 14N151 Introduction
Schur polynomials were defined by Issai Schur [12] as characters of irreducible representations of Open image in new window ; they are symmetric polynomials in n variables, indexed by Young diagrams \(\lambda \) with at most n rows. Schur polynomials form a basis in the ring of all symmetric polynomials. This basis plays a prominent role in various algebraic problems.
Schur polynomials also have a combinatorial definition: they are obtained as sums of monomials indexed by semistandard Young tableaux of shape \(\lambda \). These are fillings of boxes of \(\lambda \) by nonnegative integers satisfying certain conditions. This construction is due to W. Specht [14], who showed that such tableaux index a basis in the representation of Open image in new window corresponding to \(\lambda \).
In 1982, A. Lascoux and M.P. Schützenberger generalized the notion of Schur polynomials, defining flagged Schur polynomials. Their definition is similar to the combinatorial definition of Schur polynomials, with the only difference: they put extra constraints on the entries occuring in semistandard Young tableaux, encoded by a sequence of integers referred to as a flag. These polynomials are not symmetric anymore. However, they also satisfy an analogue of the Jacobi–Trudi identity, due to I. Gessel and M. Wachs.
Flagged Schur polynomials were defined because of their relation to Schubert polynomials. This is another remarkable family of polynomials. It is also due to A. Lascoux and M.P. Schützenberger. These polynomials are indexed by permutations; they represent cohomology classes of Schubert cycles in flag varieties. Just as Schur polynomials, they admit an algebraic definition (we recall it in Sect. 4.1), as well as a combinatorial one, due to S. Fomin and An. Kirillov [3]. The latter definition uses certain combinatorial objects called pipe dreams, or rcgraphs. It turns out that for a “nice” family of permutations, known as vexillary permutations, Schubert polynomials are equal to some flagged Schur polynomials. This gives determinantal expressions for Schubert polynomials of vexillary permutations.
In this paper we consider flagged Schur polynomials with flags of some special type (the socalled hflagged ones). We show that for these polynomials there is a formula expressing them as determinants whose entries are 1flagged Schur polynomials, divided by certain monomial denominators. This determinantal formula is different from the generalized Jacobi–Trudi formula discussed above.
The proof of our formula is based on the wellknown Lindström–Gessel–Viennot lemma. We present flagged Schur polynomials as weighted sums over tuples of nonintersecting lattice paths and express these sums as determinants using this lemma.
Then we reinterpret our formula in terms of Schubert polynomials. It expresses the Schubert polynomial of an hshifted dominant permutation as a determinant whose entries are Schubert polynomials of 1shifted dominant permutations. In particular, this gives determinantal formulas for the Schubert polynomials corresponding to the conjecturally “most singular” Schubert varieties.
This paper is organized as follows. In Sect. 2 we give the definition of flagged Schur varieties and recall the generalized Jacobi–Trudi identity. In Sect. 3 we present a path interpretation of hflagged Schur polynomials and prove our main result, Theorem 3.7. To make our exposition complete, in Sect. 3.3 we give a proof of our main combinatorial tool, the Lindström–Gessel–Viennot lemma. Section 4 is devoted to Schubert polynomials. We recall their definition, discuss their relation with flagged Schur polynomials and reformulate our main result in terms of Schubert polynomials (Theorem 4.6). Finally, in Sect. 5 we speak about its applications to some special classes of permutations and discuss its (mostly conjectural) relations to the geometry of Schubert varieties.
2 Flagged Schur polynomials and the Jacobi–Trudi identity
2.1 Definition of flagged Schur polynomials
We start with recalling the definition of flagged Schur polynomials, introduced by A. Lascoux and M.P. Schützenberger [9].
Fix two positive integers \(m\leqslant n\). Let \(\lambda =(\lambda _1\geqslant \dots \geqslant \lambda _m>0)\) be a partition and let \(b=(b_1\leqslant b_2\leqslant \dots \leqslant b_m=n)\) be a sequence of nonstrictly increasing positive integers. A flagged semistandard Young tableau T of shape \(\lambda \) and flags b is an array \(t_{ij}\) of positive integers \(t_{ij}\), \(1\leqslant j\leqslant \lambda _i\), \(1\leqslant i\leqslant m\), such that \(1\leqslant t_{ij}\leqslant t_{i,j+1}\leqslant b_i\) and \(t_{ij}<t_{i+1,j}\). In other words, the boxes of a Young diagram of shape \(\lambda \) are filled by numbers \(t_{ij}\) nonstrictly increasing along the rows and strictly increasing along the columns, in such a way that the numbers in the ith row do not exceed \(b_i\). Denote the set of all such tableaux by \({\mathscr {T}}(\lambda ,b)\). To each tableau T we assign a monomial Open image in new window , where \(p_k\) is the number of entries in T equal to k.
Definition 2.1
Example 2.2
If \(b_1=\dots =b_m=n\), then \({\mathscr {T}}(\lambda ,b)\) is the set of semistandard Young tableaux of shape \(\lambda \) whose entries do not exceed n. In this case \(s_\lambda (b)\) is just the usual Schur polynomial \(s_\lambda (x_1,\dots ,x_n)\) in n variables.
2.2 hflagged Schur polynomials
We will be mostly interested in flagged Schur polynomials with flags of a special form.
Definition 2.3
Let h be a positive integer, \(b=(h+1,h+2,\dots , h+m)\). Then \(s_{\lambda }^{(h)}=s_\lambda (b)\) is called an hflagged Schur polynomial of shape \(\lambda \). We will also speak about hflagged semistandard Young tableaux of shape \(\lambda \).
Example 2.4
1flagged semistandard Young tableaux of shape \(\lambda \) correspond to Young subdiagrams \(\mu \subset \lambda \). Indeed, let T be such a tableau. The entries \(t_{ij}\) in each (ith) row of T are equal either to i or to \(i+1\). The union of all boxes \((i,j)\in T\) such that \(t_{ij}=i\) forms a subdiagram \(\mu \), and \(\mu \) uniquely determines the tableau T.
Example 2.5
Example 2.6
2.3 Generalized Jacobi–Trudi identity
The following determinantal expression for flagged Schubert polynomials is well known. It appeared first in the paper [5] by I. Gessel. Its proof is also given in M. Wachs’s paper [17, Theorem 1.3]. In their paper [2], Y. Chen, B. Li and J. Louck generalize this identity for the case of double flagged Schur polynomials and give yet another its proof, using the lattice path interpretation of these polynomials and the Lindström–Gessel–Viennot lemma.
Theorem 2.7
Corollary 2.8
Remark 2.9
The “path interpretation” of the Jacobi–Trudi formula and its proof using the Lindström–Gessel–Viennot lemma can be found, for instance, in [15, 7.16].
3 Another determinantal formula for flagged Schur polynomials
3.1 Path interpretation of 1flagged Schur polynomials
Let \(\lambda =(\lambda _1,\dots ,\lambda _{n})\) be a fixed partition. Consider an oriented graph \(G_\lambda \) whose vertices are lattice points inside \(\lambda \) or at its boundary, and edges are given by vertical and horizontal line segments joining the neighboring lattice points. The orientation on the edges is as follows: the horizontal and the vertical edges are directed towards east and north, respectively. Let us assign weights to the edges of \(G_\lambda \) in the following way: to each horizontal edge \((r,s)(r+1,s)\) we assign the weight \(x^{1}_{s+1}\), as shown on Fig. 1, while all vertical edges have weight 1. For a path P, its weight \(\mathop {\mathrm {wt}}P\) is defined as the product of the weights of all its edges.
Definition 3.1
The following easy proposition expresses 1flagged Schur polynomials via partition functions.
Proposition 3.2
Proof
Example 3.3
3.2 Expressing hflagged Schur polynomials via 1flagged Schur polynomials
In this subsection we give a lattice path expression for hflagged Schur polynomials, different from the one given in Theorem 2.7. Using the Lindström–Gessel–Viennot lemma, we express them as determinants involving 1flagged polynomials from the previous subsection.
As before, let \(\lambda =(\lambda _1,\dots ,\lambda _m)\) be a Young tableau, and let \(h\geqslant 1\). We consider hflagged Young tableaux: the tableaux with flags \((h+1,\dots ,h+m)\). To each such tableau T we can assign an htuple of lattice paths drawn in the fourth quarter of the plane that start at the point Open image in new window , end at \(B=(\ell ,0)\), where \(\ell =\lambda _1\), and pass inside the diagram \(\lambda \). These paths are constructed in the following way. For \(k\leqslant h\), denote by \(\mu _k\) the set of boxes \(t_{ij}\) of T such that \(t_{ij}\leqslant i+k1\). We thus obtain a set of embedded Young tableaux \(\mu _1\subset \mu _2\subset \cdots \subset \mu _h\subset \lambda \). Consider the set of lattice paths \(P_1,\dots ,P_h\) bounding these diagrams; they pass nonstrictly above one another, and all of them are bounded by \(\lambda \) from below.
Example 3.4
Fig. 2 shows a 2flagged Young tableau for the diagram \(\lambda =(5,4,4,2)\). It corresponds to the pair of paths, shown at the figure on the right.
Now let us shift each (kth) path to the vector Open image in new window . This will give us an htuple of noncrossing paths. The kth path \(P_k\) will then start at the point Open image in new window and end at Open image in new window .
Let us also extend paths as shown on the picture: the kth path \(P_k\) is extended by \(hk\) vertical segments down and by \(hk\) horizontal segments to the right. Denote the extended path by \(P_k'\) and its endpoints by \(A_k'\) and \(B_k'\). The extensions are shown on Fig. 3 by dashed lines.
Definition 3.5
Let \(\lambda =(\lambda _1,\dots ,\lambda _m)\) be a Young diagram, and \(k,\ell \) two nonnegative integers. Let us shift \(\lambda \) by the vector Open image in new window . The minimal Young diagram containing this set is denoted by \(\lambda (k,\ell )\). In other terms, Open image in new window is obtained from \(\lambda \) by adding k rows of length \(\lambda _1\) from above and \(\ell \) columns of height m from the left.
Lemma 3.6
This lemma together with the expressions for Schur polynomials via lattice paths gives us the following result which expresses an hflagged Schur polynomial via 1flagged Schur polynomials.
Theorem 3.7
Remark 3.8
This determinantal formula is different from the Jacobi–Trudi formula (and, to the best of our knowledge, cannot be reduced to it). Note, in particular, that all matrix entries in the Jacobi–Trudi formula are polynomials in \(x_1,\dots ,x_n\), while the matrix elements in our formula are polynomials in different sets of variables.
Example 3.9
We will also need a slightly different form of this determinantal formula. It can be obtained from Theorem 3.7 by algebraic manipulations with determinants, but we will give a combinatorial proof instead. Let us modify Definition 3.5 in the following way.
Definition 3.10
In other words, Open image in new window is obtained from \(\lambda \) by adding “staircases” of size k and \(\ell \) on the top and to the left of it, respectively.
Example 3.11
Note that the kth path is situated inside the diagram Open image in new window . This allows us to formulate the following corollary of Theorem 3.7. Its proof also follows from the Lindström–Gessel–Viennot lemma; the only difference concerns the weights which are added while extending paths.
Corollary 3.12
Remark 3.13
This corollary can also be obtained from Theorem 3.7 by purely algebraic manipulations with the determinant.
Remark 3.14
Note that the righthand side depends upon \(x_k\) for \(k>h+m\), while the lefthand side does not. This means that all these \(x_k\) cancel out while evaluating the determinant.
We will need this form of our determinantal identity while dealing with Schubert polynomials in the next section.
3.3 The Lindström–Gessel–Viennot lemma
In this subsection we prove the Lindström–Gessel–Viennot lemma on noncrossing paths. This material is by no means new: having appeared first in the preprint [6], now it can be found in many books and articles, for instance, in [16].
Lemma 3.15
Proof
This map on the htuples of crossing paths is involutive. Obviously, \(\mathop {\mathrm {wt}}P_i\mathop {\mathrm {wt}}P_j=\mathop {\mathrm {wt}}\widetilde{P}_i\mathop {\mathrm {wt}}\widetilde{P}_j\), so it is weightpreserving. Finally, the permutations corresponding to \((P_1,\dots ,P_h)\) and \((P_1,\dots ,\widetilde{P}_i,\dots ,\widetilde{P}_j,\dots ,P_h)\) are obtained from each other by multiplication by a transposition, so they are counted in expression (1) with the opposite signs. This means that the total contribution of this pair of htuples of paths into (1) is zero, and the determinant is equal to the sum of weights over all htuples of noncrossing paths. The lemma is proven. \(\square \)
4 Schubert polynomials
4.1 Definition of Schubert polynomials
Schubert polynomials were introduced for studying the cohomology ring of a full flag variety. Their definition is due to A. Lascoux and M.P. Schützeberger [9]; implicitly it appeared in the paper [1] by J. Bernstein, I. Gelfand and S. Gelfand. Let us recall it.
Let \(S_n\) denote the symmetric group, i.e., the group of permutations of an nelement set \(\{1,\dots , n\}\). We will use the oneline notation for permutations; i.e., (1342) is a permutation taking 1 to 1, 2 to 3, 3 to 4 and 4 to 2. Let \(s_i=(1,2,\dots ,i+1,i,i+2,\dots ,n)\) denote the ith simple transposition, and \(w_0=(n,n1,\dots ,2,1)\) the element of maximal length. We mean by \(\ell (w)\) the length of the permutation w, i.e., the minimal number k of simple transpositions \(s_{i_1},\dots ,s_{i_k}\) such that \(w=s_{i_1}\!\dots s_{i_k}\).
Definition 4.1
There are standard embeddings \(S_n\hookrightarrow S_m\) for \(n<m\); under this embedding, \(S_n\) permutes only the first n elements of \(\{1,\dots , m\}\). Sometimes we will treat permutations as elements of the direct limit \(S_\infty =\lim _{\rightarrow } S_n\). It can be shown (cf. [10]) that the Schubert polynomials are stable under these embeddings, so they can be regarded as elements of the polynomial ring Open image in new window in countably many variables.
4.2 Schubert and flagged Schur polynomials
Recall that a permutation \(w\in S_n\) is said to be vexillary if it is (2143)avoiding, i.e., there are no four numbers \(i<j<k<\ell \) such that \(w(j)<w(i)<w(\ell )<w(k)\). M. Wachs [17] shows that the Schubert polynomials of vexillary permutations can be obtained as flagged Schur polynomials for certain flags. This is done as follows.
We define the inversion sets of a permutation w as \({{\mathrm{I}}}_i(w)=\{j: j>i,\, w(j)<w(i)\}\). It can be shown that a permutation is vexillary iff its inversion sets form a chain (see [11] for details).
Let \(c_i=\#\, {{\mathrm{I}}}_i\). The sequence of these cardinalities \((c_1,\dots ,c_n)\) is called the Lehmer code of w. Denote by \(\lambda (w)\) the partition obtained by arranging \(c_1,\dots , c_n\) in the decreasing order. Likewise, let \(b(w)=(b_1(w)\leqslant \cdots \leqslant b_m(w))\) be the sequence of integers \(\min {{\mathrm{I}}}_i(w)1\) for all nonempty \({\mathrm{I}}_i(w)\), arranged in the increasing order.
Theorem 4.2
([17, Theorem 2.3]) If w is vexillary, then \({\mathfrak {S}}_w=s_{\lambda (w)}(b(w))\).
4.3 Dominant permutations
Definition 4.3
A permutation \(w\in S_n\) is called dominant if it is 132avoiding, i.e., there is no triple \(i<j<k\) such that \(w(i)<w(k)<w(j)\).
Clearly, if a permutation is 132avoiding, it is also 2143avoiding. So, all dominant permutations are vexillary.
It is well known (cf., for instance, [11]) that dominant permutations are characterized by the following property: their Schubert polynomial \({\mathfrak {S}}_w\) is a monomial in \(x_1,\dots ,x_{n1}\). Moreover, the Lehmer code \(\lambda (w)\) of such a permutation is a partition, and \({\mathfrak {S}}_w=\underline{(x_1,\dots ,x_{n1})}^{\lambda (w)}\).
A flagged tableau of shape \(\lambda \) with this flag is unique: all its entries in the ith row are equal to i. So we can replace this flag by \((1,2,3,\dots )\). Hence, for a dominant permutation w its Schubert polynomial is a “0flagged Schur polynomial of shape \(\lambda \)”.
4.4 Shifts of permutations
Proposition 4.4

Open image in new window is vexillary;
This gives us a description of permutations whose Schubert polynomials are equal to hflagged Schur polynomials.
Remark 4.5
Of course, for a dominant permutation w, the permutations Open image in new window are not dominant anymore.
4.5 Extension of dominant permutations
In this subsection we define an operation on dominant permutations which corresponds to \((k,\ell )\)extension of their Young diagrams.
A \((k,\ell )\)extension Open image in new window of a dominant permutation is obtained as a result of k successive applications of (1, 0)extension and \(\ell \) applications of (0, 1)extension (these operations commute). Again, the Young diagram corresponding to its code is obtained from the Young diagram of w by a \((k,\ell )\)extension.
4.6 Determinantal formula for Schubert polynomials of shifted dominant permutations
With the definitions from the two previous subsections, we can reformulate Corollary 3.12 for Schubert polynomials. The formula from that theorem will give the determinantal expression for the Schubert polynomial of an hshifted dominant permutation w in terms of 1shifted dominant permutations obtained from w by extensions.
Theorem 4.6
5 Schubert polynomials of Richardson permutations
In this section we discuss some corollaries of Theorem 4.6 and formulate several conjectures concerning Schubert polynomials.
5.1 Catalan numbers
Fix some notation. Let \(w_0(n)=(n,n1,\dots ,2,1)\in S_n\) be the element of maximal length. It is dominant, and the corresponding Young diagram is the staircase shape \({\Lambda }_n=(n1,n2,\dots ,2,1)\).
Theorem 5.1
Corollary 5.2
In this case \({\mathfrak {S}}_w=s_{{\Lambda }_n}^{(1)}\) is the 1flagged polynomial of the staircase partition.
5.2 Catalan–Hankel determinants
Consider the same permutation \(w_0(n)\) shifted by h instead of 1. Theorem 4.6 expresses \({\mathfrak {S}}_{1^h\times w_0}\) via \({\mathfrak {S}}_{1\times \widehat{w_0(n)}[k,\ell ]}\).
Note that Open image in new window , since the extension of a staircase diagram is again a staircase diagram. This means that all permutations occurring in the determinantal expression are \({\mathfrak {S}}_{1\times w_0(m)}\) for \(m=n,\dots ,n+2h2\).
Specializing this identity at 1 and using Corollary 5.2, we get the following proposition.
Proposition 5.3
This determinant is a Hankel determinant of the sequence of Catalan numbers; we will later refer to it as a Catalan–Hankel determinant.
Remark 5.4
Since \({\mathfrak {S}}_{1^h\times w_0(n)}\) is the flagged Schur polynomial \(s_{{\Lambda }_n}^{(h)}\), we see from the definition of a flagged Schur polynomial that its value at 1 is equal to the number of plane partitions with parts at most h and shape \({\Lambda }_n\), i.e., threedimensional Young diagrams inside a prism with base \({\Lambda }_n\) and height h. This observation appeared in [4, Section 1.2].
It is also possible to establish an explicit bijection between the monomials in \({\mathfrak {S}}_{1^h\times w_0(h)}\), indexed by the socalled “pipe dreams” (see [3]), and such plane partitions. See [13] for the details.
5.3 Richardson elements in the symmetric group
Definition 5.5
Let \(i_1<i_2<i_3<\dots <i_k=n\). Permutation \((i_1,\,i_11,\) \(\dots ,\,2,\,1,\,i_2,\,i_21,\,\dots ,\,i_1+1,\, i_3,\,\dots ,\,i_2+1,\,\dots )\in S_n\) is called a Richardson permutation.
In different terms, a Richardson permutation is equal to the product of several permutations of type Open image in new window with nonintersecting supports. In particular, \(w_0(n)\) and Open image in new window are Richardson permutations.
Richardson elements have a nice description in terms of algebraic groups. For a parabolic subgroup Open image in new window consider its Levi decomposition Open image in new window . Then the longest element in the Weyl group \(W_L\) of L is a Richardson element in the Weyl group \(W\cong S_n\) of Open image in new window .
Richardson permutations are not vexillary (unless they are shifted staircase permutations). However, they are obtained as products of several shifted staircase permutations with pairwise disjoint supports. So the Schubert polynomial of such a permutation is equal to the product of the Schubert polynomials of its factors.
Hence, Theorem 4.6 shows that the Schubert polynomials of Richardson permutations can be presented as products of several determinants, and their specialization at 1 is equal to the product of several Catalan–Hankel determinants from Proposition 5.3.
The study of Richardson permutations is motivated by the following question.
Question 5.6
For a given n, find the permutations \(w\in S_n\) such that \({\mathfrak {S}}_{w}(1,\dots ,1)\) is as large as possible.
Permutations producing the largest values of \({\mathfrak {S}}_w(1,\dots ,1)\)
n  w  \({\mathfrak {S}}_w(1,\dots ,1)\) 

2  (12)  1 
3  (132)  2 
4  (1432)  5 
5  (15432)  14 
(12543)  14  
(21543)  14  
6  (126543)  84 
(216543)  84  
7  (1327654)  660 
8  (13287654)  9438 
9  (132987654)  163592 
10  (1,4,3,2,10,9,8,7,6,5)  4424420 
These values were computed for all permutations with \(n\leqslant 10\). It turned out that the maximum is always attained at a Richardson permutation. This allows us to formulate the following conjecture.
Conjecture 5.7
For a given n, the permutation \(w\in S_n\) with the largest value of \({\mathfrak {S}}_w(1,\dots ,1)\) is Richardson.
In Table 1 we give the list of permutations w corresponding to the largest value of \({\mathfrak {S}}_w(1,\dots ,1)\) and the values themselves for \(n\leqslant 10\) (for \(n=5,6\), there is more than one such permutation). This table was computed by I. Kochulin [8].
However, this conjecture remains a purely experimental observation; so far we cannot see any reason why the permutation producing the largest value of its Schubert polynomial at 1 has to be Richardson.
Notes
Acknowledgments
This paper was completed during the second author’s visit to the University of Warwick; he is grateful to this institution and to Professor Miles Reid for their warm hospitality.
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