For any flag, the rank matrix is the $ (n+1)\times(n+1)$ matrix $ r=(r_{ij})$ with the integer entries \begin{eqnarray*} r_{ij}=\dim{}V_i\cap\Csps^j, \quad 0\leq i, j\leq{}n. \end{eqnarray*} The rank matrix is independent of the choice of a flag in a $ B$-orbit. Moreover, it completely characterizes the corresponding $ B$-orbit. More precisely, two different flags, $ F\in{}{\mathcal C}_w$ and $ F'\in{}{\mathcal C}_{w'}$, have the same rank matrix if and only if $ w=w'$; see, e.g., [Fulton1997]. We will denote by $ r(w)$ the rank matrix corresponding to the Schubert cell $ {\mathcal C}_w$.

Obviously, one has: \begin{eqnarray*} \begin{array}{l} r_{0k}=r_{k0}=0; \qquad r_{kn}=r_{nk}=k;\\ r_{ij}+r_{i+1,j+1}\geq r_{i+1,j}+r_{i,j+1};\\ r_{i,j+1}- r_{ij}=0\quad\hbox{or}\quad1;\\ r_{i+1,j}- r_{ij}=0\quad\hbox{or}\quad1. \end{array} \end{eqnarray*} Every integer matrix $ (r_{ij})$ with the above properties is the rank matrix of some flag.

The following statement is due to [Fulton1992], see also [Fulton1997] p. 157. The Schubert cell $ {\mathcal C}_{w}$ consists in flags such that the corresponding rank matrix is:

The permutation $ w\in{}S_n$ can be easily recovered from the rank matrix.

To make this visible, we usually put a $ \bullet$ into the matrix, so that the permutation is encoded by the dots.

Proof.

The smaller is the Schubert cell $ {\mathcal C}_w$, the bigger are the numbers $ r_{ij}(w)$.

The rank matrix is completely determined by a few particular entries. This idea is due to [Fulton1997] (see also [Woo2009], [Reiner et al.2011] and references therein). The following notion is crucial for us.

We always encircle the pillar entries, in order to distinguish them.

In combinatorial terms, pillar entries can be characterized as follows. An entry $ r_{ij}$ of a rank matrix $ r(w)$ is pillar if and only if

It worth noticing that, the more a given permutation $ w$ is "close" to the identity, the more pillar entries the matrix $ r(w)$ has. The matrix $ r(\Id)$ has $ n-1$ pillar entries $ r_{ii}=i$, for $ 1\leq{}i\leq{}n-1$. The more $ w$ is "close" to the longest element $ w_0$, the less pillar entries the matrix $ r(w)$ has. In particular, $ r(w_0)$ is the only rank matrix with no pillar entries.

This statement is classical. For the sake of completeness, a proof will be presented in Sect. 5.6. An explicit algorithm that reconstructs the permutation $ w$ from the pillar entries of the rank matrix $ r(w)$ will be presented in Sect. 4.2.

Let us describe the pillar entries of the rank matrices corresponding to the Coxeter elements.

Proof.

We believe that the notion of pillar entry deserve a further study. In particular, the number of pillar entries for a given permutation is an interesting characteristic. Some of the basic properties of pillar entries will be presented in Sect. 4.

Let us recall here Fulton's notion of essential entry. An entry $ r_{ij}$ of a rank matrix $ r(w)$ is called essential , see [Fulton1992] and also [Eriksson and Linusson1996], if

It is proved in [Fulton1992] that every rank matrix (and therefore the corresponding Schubert variety) is completely characterized by its essential set.

The notions of essential and pillar entries are somewhat "complementary", as the inequality signs in formulas (2.3) and (2.4) are reversed, cf. Appendix for a comparison.

The following conjecture asserts that if two Schubert varieties have the same tangent cones, then they have the same number of pillars, whose values are also the same, and whose position in the respective rank matrices can only differ by transposition.

Example 2.1 and Proposition 2.8 are the first examples that confirm our conjecture. We will give many other examples in the sequel.

Note that the inverse of Conjecture 2.10 is false: two permutations with partially transposed pillar entries do not necessarily correspond to the same tangent cones.

Figure 1 Permutations of different length with transposed pillars.

Another restriction for partial transposition of pillars occurs more often than the above discussed one. Given a permutation $ w$ and the corresponding rank matrix $ r(w)$, then a partial transposition of the pillar entries may not correspond to any rank matrix of any permutation.

Example 2.12.

Consider the permutation $ w=34521$ in $ S_5$. The corresponding rank matrix is as follows:

It turns out that there are no rank matrices with the following pillar entries:

Indeed, the above positions of pillar entries are impossible, since they contradict formula (2.3), see also Sect. 4.1 for more details.

Let us briefly discuss the partial transpositions of linked pillar entries. If one transpose some pillar entries of a rank matrix $ r(w)$, but not all of them, then the following three possibilities may occur:

1. (1) there exists a rank matrix of a permutation $ w'$ that does have the given set of pillar entries, but of different length (cf. Example 2.11);
2. (2) there is no rank matrix of a permutation that has this set of pillar entries (cf. Example 2.12);
3. (3) the "good case" where the resulting matrix is a rank matrix of a permutation that has the given set of pillar entries and the same tangent cone as $ w$.

We define an equivalence relation on the set of pillar entries of a rank matrix. Roughly speaking, two pillar entries are in the same class if they are "close enough" to each other.

[]

Figure 2 Configurations for two related pillars.

Figure 3 Configurations for two dissociated pillars .

Figure 4 Permutation with three classes of linked pillars.

It is convenient to display the linking relations between the pillar entries using a graph.

For instance, Example 3.2 corresponds to the following graph (Fig. 5)

Figure 5 Graph of related pillars .

The connected components of the linking graph correspond to the classes of linked pillar entries.

An admissible partial transposition is an operation defined on rank matrices and on the group $ S_n$. Roughly speaking, it consists in transposition of a part of the pillar entries, such that linked pillar entries transpose (or not) simultaneously. More precisely, we have the following:

Figure 6 An example of admissible partial transposition.

In this section we formulate a sufficient condition for the tangent cones of two Schubert varieties to coincide. Furthermore, it turns out that every partial transposition of the pillar entries in the associated rank matrices defines an operation on the group $ S_n$.

Our main result is the following

We will prove this theorem in Sects. 4.5 and 5.7.

Proof.

We will give an explicit description of the corresponding tangent cone in Sect. 5.3. Note also that the Schubert varieties corresponding to the Coxeter elements are smooth. Therefore, Corollary 3.8 can also be deduced from the theorem of Lakshmibai, see [Lakshmibai1995] that describes the Zariski tangent space.

Theorem 3.6 provides large classes of Schubert varieties with identical tangent cones. However, these classes can be yet larger. In fact, there are other cases of partial transposition of pillar entries than those considered above.

Figure 7 An example of admissible transposition of linked pillars.

This example is not covered by Theorem 3.6 and shows its limits. For instance, it shows that the converse statement to Part (ii) of the theorem is false. Existence of such partial transpositions of pillar entries constitutes the main difficulty in solving the initial classification problem.

The rank matrix $ r(w)$ is determined by the diagram of the corresponding permutation $ w$.

Proof.

The positions of the pillar entries in $ r(w)$ can be determined by local structure of the diagram of $ w$. Consider horizontal strips of height $ 1$ and a vertical strips of width $ 1$ in the diagram, such that the upper left and the lower right corners are marked dots of the permutation:

Proof.

Figure 8 Localizing the pillars.

It will be useful in the sequel to have the following observation.

Proof.

  Let $ (i,j),\;(i+1,j+k)$, $ k> 0$, be the marked dots of a horizontal strip of height one. If $ k=1$, then our horizontal strip is also a vertical strip, and these two (identical) strips intersect each other. Let $ k> 1$, and let $ (i_1,j+1),\,\ldots,\,(i_{k-1},j+k-1)$ be the marked dots of the diagram of $ w$, lying on the vertical lines crossing our strip; neither of $ i_1,\,\ldots,i_{k-1}$ is $ i$ or $ i+1$. If $ i_1> i+1$, then the vertical strip with marked dots $ (i,j),\,(i_1,j+1)$ intersects our horizontal strip. Similarly, if $ i_{k-1}< i$, then the vertical strip with marked dots $ (i_{k-1},j+k-1),\;(i+1,j+k)$ intersects our horizontal strip. If $ i_1< i$ and $ i_{k-1}> i+1$, then for some $ s$, $ i_s< i$ and $ i_{s+1}> i+1$. In this case, the vertical strip with marked dots $ (i_s,j+s),\,\ldots,\,(i_{s+1},j+s+1)$ intersects our horizontal strip.

The proof of the vice-versa statement is the same, with the coordinates of every marked dot switched. ⬜

In this section we present an algorithm of constructing the diagram of $ w$ from the set of pillar entries.

Let us introduce some useful notation. First we numerate the pillar entries in the lexicographical order, that is, from left to right in each row and then counting the rows from top to bottom. Then we set: $ (p_i,q_i)=$ the position of the $ i$-th pillar; $ K_i=r_{p_iq_i}(w)=$ the value of the $ i$-th pillar entry; $ NW_i=$ The North-west region of the $ i$-th pillar entry.

We draw the $ (n+1)\times(n+1)$ square grid; columns and rows of this grid are separated by $ n$ horizontal lines and $ n$ vertical lines. We mark the given pillar entries in $ N$ cells of the grid (thus, $ N$ is the number of pillar entries). Our permutation $ w$ will appear as a set of $ n$ dots in the intersections of horizontal and vertical lines, one on each horizontal line and one on each vertical line.

The diagram of $ w$ is constructed in $ N+1$ steps. At every step, we place some dots into the interior of the region $ NW_i$. If the action requested at any step is impossible by any reason, then our set of "pillar entries" is not the set of pillar entries of $ r(w)$ for any $ w$.

For $ i=1,\dots,N$, at the $ i$-th step, we fist count the number of dots placed in the interior of $ NW_i$ at the previous steps. If this number is $ L$, we need to add $ k_i=K_i-L$ dots into \begin{eqnarray*} NW_i-(NW_1\cup\dots\cup NW_{i-1}). \end{eqnarray*} For this, we numerate the horizontal and vertical lines within $ NW_i$ which do not bear any of the $ L$ dots placed at the previous steps, respectively from bottom to tor and from right to left. Then, for $ j=1,\dots,k_i$ we place a dot at the intersections of the vertical line number $ j$ and the horizontal line number $ k_i+1-j$ (Fig. 9).

Figure 9 Placement of the dots.

Our algorithm requests that neither of these dots falls into any of the regions $ NW_1,\dots,NW_{i-1}$.

The final, $ (N+1)$-st step works according the same rules with the whole matrix playing the role of $ NW_{N+1}$.

The above algorithm is the only way to mark dots without creating an extra pillar or changing the values of the pillar entries. Note that, for the Fulton essential set, a reconstruction algorithm is given in [Eriksson and Linusson1996].

Figure 10 Recovering the permutation $ w$ from the pillar entries.

The dimension and codimension of a Schubert cell $ {\mathcal C}_w$ (or a Schubert variety $ \X_w$) can be computed directly form the set of pillar entries of the corresponding rank matrix $ r(w)$.

The number \begin{eqnarray*} \codim({{\mathcal C}_w})=\ell(ww_0)=\#\{i< j : w(i)< w(j)\} \end{eqnarray*} can be obtained in the diagram of $ w$ counting the intersections of the horizontal segments and the vertical segments of the grid that are at the right and above each dots, respectively:

Figure 11 $ \ell(ww_0)$ from the diagram of $ w$.

The following formula gives the codimension of a Schubert cell from the data of its pillar entries.

Proof.

Given a permutation of $ w\in{}S_{n}$, we will show the existence of permutations whose pillar entries form the subsets in the set of pillar entries of $ r(w)$ obtained by removing of some classes of linked pillar entries.

The pillar entries of $ r(w)$ are decomposed in the disjoint union of classes of linked pillar entries: $ \{r_{ij}(w)\}=\Lc_{1}\sqcup \Lc_{2}\sqcup \ldots \sqcup \Lc_{s}.$ These classes correspond to subintervals $ I_1,I_2,\ldots,I_s$ of the interval $ [0,n]$; these subintervals have integer endpoints and pairwise have no interior points. The class $ \Lc_t$ corresponds to the interval $ I_t$, if, for every $ r_{ij}(w)\in\Lc_t$ both $ i$ and $ j$ belong to $ I_t$. We order the intervals $ I_t$, from the left to the right, and order the classes $ \Lc_t$ accordingly. Consequently, if $ u< v$ and $ r_{ij}(w)\in \Lc_{u},\,r_{i'j'}(w)\in \Lc_{v}$, then $ i\leq{}i',\,i\leq{}j',j\leq{}i',\,j\leq{}j'$. In particular, for $ u< v$, all pillar entries from $ \Lc_{u}$ lexicographically precede the pillar entries from $ \Lc_{v}$.

The following statement is our first application of the reconstruction algorithm presented in Sect. 4.2.

Proof.

The lexicographic order suggests a natural series of admissible partial transpositions, such that all the classes of linked pillar entries $ \Lc_i$ transpose for $ i$ less or equal to some value. In this section we present en explicit algorithm of calculating the resulting permutations. This algorithm is the main ingredient of the proof of Part (i) of Theorem 3.6.

For $ t\in \{1, \ldots, s\}$, we define the elementary partial transposition $ w'=\trp_{t}(w)$, as the permutation having the following set of pillar entries: \begin{eqnarray*} \left\{ \begin{array}{rclll} r_{ji}(w')&=&r_{ij}(w), & \text{if} &r_{ij}(w) \in \Lc_{1}\sqcup \ldots \sqcup \Lc_{t};\\ r_{ij}(w')&=&r_{ij}(w), & \text{if} &r_{ij}(w) \in \Lc_{t+1}\sqcup \ldots \sqcup \Lc_{s}.\\ \end{array} \right. \end{eqnarray*} Note that every partial transposition can be obtained as a sequence of elementary partial transpositions.

Given a permutation $ w=w_1w_2\ldots{}w_n\in{}S_n$, the entries $ w_k$ of $ w$ are separated into two disjoint groups, $ I_1\sqcup{}I_2$: \begin{eqnarray*} \left\{ \begin{array}{rclll} w_k\in{}I_1,& \text{if}&k\leq{}\max(j), & \text{for pillars} &r_{ij}(w) \in \Lc_{1}\sqcup \ldots \sqcup \Lc_{t};\\ w_k\in{}I_2,& \text{if}&k> {}\min(i), & \text{for pillars} &r_{ij}(w) \in \Lc_{t+1}\sqcup \ldots \sqcup \Lc_{s}.\\ \end{array} \right. \end{eqnarray*} The algorithm of calculation the permutation $ w'=w'_1w'_2\ldots{}w'_n$, obtained via the above elementary partial transposition, consists in three steps:

1. (1) keep $ w'_k=w_k\in{}I_1$ if $ w_k\leq{}k$, and $ w'_k=w_k\in{}I_2$ if $ w_k\geq{}k$;
2. (2) inverse the entries $ w_k\in{}I_1$, i.e., write $ k$ at position $ w_k$;
3. (3) fill the remaining positions in $ w'$ in the decreasing order.

The proof of the above algorithm is straightforward.

Example 4.11.

For the Coxeter element $ w=2341\in{}S_4$, the elementary transposition

is obtained into three steps: \begin{eqnarray*} 2\,3\,\vert\,4\,1 \quad\to\quad 2\,.\,\vert\,4\,. \quad\to\quad .\,1\,\vert\,4\,. \quad\to\quad 3\,1\,\vert\,4\,2, \end{eqnarray*} so that $ w'=3142$ is another Coxeter element, already considered in Example 2.4, b).

For every $ w\in{}S_n$, the above algorithm implies the existence of a permutation $ w'$ such that the pillar entries of $ r(w')$ are obtained by an admissible partial transposition of pillar entries of $ r(w)$.

Part (i) of Theorem 3.6 is proved.

In the neighborhood of the standard flag $ F_0$, the flag variety $ \F$ is identified with the subgroup of unitriangular matrices

Our next goal is to describe the Schubert cells and Schubert varieties in terms of this coordinate system.

Let $ M_{ij}$ be the $ (n-j)\times{}i$ submatrix of $ X$ consisting of the last $ n-j$ rows and the first $ i$ columns.

1. (a) If $ i\leq{}j$, then this submatrix is of the form \begin{eqnarray*} M_{ij}= \left( \begin{array}{ccc} x_{j+1\,1}&\cdots&x_{j+1\,i}\\ \vdots&&\vdots\\ x_{n1}&\cdots&x_{ni} \end{array} \right). \end{eqnarray*}
2. (b) If $ i> j$, then the submatrix is as follows \begin{eqnarray*} M_{ij}= \left( \begin{array}{llcc} x_{j+1\,1}\cdots\; x_{j+1\,j}&1&\\ \vdots&\;\;\;\;\ddots\\ x_{i1}\;\;\;\;\;\cdots&x_{i\,i-1}&1\\ \vdots&&\vdots\\ x_{n1}\;\;\;\;\;\cdots&\cdots&x_{ni} \end{array} \right). \end{eqnarray*}

The following lemma translates the description of Schubert cells in terms of rank matrices into an algebraic descripition in the above coordinate system.

Proof.

The Schubert variety $ \X_w$ is determined, in a neighborhood of the standard flag $ F_0$, by a system of polynomial equations in the variables $ x_{ij}$. The equations are obtained as follows. For each couple of indices $ i, j$, formula (5.2) leads to a set of equations that expresses the annihilation of the minors of the matrix $ M_{ij}$ of size larger than its rank. From Proposition 2.7, it suffices to consider only the equations for the indices $ i,j$ corresponding to a pillar entry $ r_{ij}(w)$ in the rank matrix of $ \X_w$.

The system of equations of the tangent cone $ \T_w$ of $ \X_{w}$ is obtained, roughly speaking, as the homogeneous lower degree parts of the equations of $ \X_w$. More precisely, the equations of $ \X_w$ can be written in such a way that the homogeneous terms of lower degree are linearly independent. Then the system of $ \T_w$ is obtained by removing all of the monomials of higher degree in the equations of $ \X_w$.

Example 5.2.

As we mentioned in Introduction, the first example of a Schubert variety with singularity at the origin correspond to the permutation $ w=4231\in{}S_{4}$ (see [Billey and Lakshmibai2000], [Lakshmibai and Sandhya1990]). Written in our local coordinates: \begin{eqnarray*} \left( \begin{array}{llll} 1\\ x_{21}&1\\ x_{31}&x_{32}&1\\ x_{41}&x_{42}&x_{43}&1 \end{array} \right) \end{eqnarray*} the equation of the corresponding tangent cone $ \T_{w}$ (the same as the equation of $ \X_{w}$) is: $ x_{31}x_{42}-x_{32}x_{41}=0$. Indeed, the rank matrix of $ w$ is as follows:

so that the Schubert cell $ {\mathcal C}_{w}$ is determined by the condition $ \dim(V_{2}\cap{}\Csps^{2})=1$, that translates in coordinates as the condition that a certain linear combination of two first column vectors belong to the subspace $ \Csps^{2}$, i.e., the matrix $ M_{22}$ degenerates.

The tangent cone $ \T_{w}$ is $ 5$-dimensional, whereas the Zariski tangent space is the whole $ 6$-dimensional tangent space $ T_{F_{0}}\F$.

In the case $ i\leq{}j$, the minors of $ M_{ij}$ are homogeneous polynomial expressions. The following observation explains the reason for which two pillar entries transposed to each other, in many situation give the same contribution to the system of equation of the tangent cones.

If $ i> j$, then $ M_{ji}$ is the complement of the upper right square submatrix in $ M_{ij}$ (of size $ i-j$) with $ 1$'s on the diagonal: \begin{eqnarray*} M_{ij}= \left( \begin{array}{ll|ccc} &&1&\\ &&\vdots&\ddots\\ &&&\cdots&1\\ \hline &M_{ji}\\ \end{array} \right). \end{eqnarray*} The lower degree homogeneous part in the expression of any minors of $ M_{ij}$ of size $ r\geq i-j$ involving the last $ i-j$ columns corresponds precisely to a minor of $ M_{ji}$ of size $ r-i+j$, and vice versa.

Let us show that the pillar entries determine the rank matrix. We use the fact that the rank matrix $ r(w)$ completely determines the Schubert variety $ \X_{w}$.

For a permutation $ w$, let $ r_{i_1j_1},\ldots,r_{i_Nj_N}$ be the pillar entries of the matrix $ r(w)$ and let $ C_{ij}$ be the condition \begin{eqnarray*} \rank(M_{ij})\le i-r_{ij}(w) \end{eqnarray*} from the system of conditions determining the Schubert variety $ \X_{w}$ (see Sect. 5.3). There are obvious implications: If $ r_{i+1,j}=r_{ij}$, then $ C_{ij}$ implies $ C_{i+1,j}$; If $ r_{i+1,j}=r_{ij}+1$, then $ C_{i+1,j}$ implies $ C_{ij}$; If $ r_{i,j+1}=r_{ij}$, then $ C_{i,j}$ implies $ C_{i,j+1}$; If $ r_{i,j+1}=r_{ij}+1$, then $ C_{i,j+1}$ implies $ C_{ij}$.

Let us visualize the matrix $ [C_{ij}]$ as an $ (n+1)\times(n+1)$ grid and show the above implications by arrows between the neighboring cells; the resulting diagram for the matrix from Example 2.12 is shown below (Fig. 12).

Figure 12 The grid of implications.

Take any cell of the grid and trace a path from it in the following way: we move in the direction opposite to the arrow. If there arises a choice of several such direction, choose any of them. If there is no such directions, then stop. It is important that our path never passes through a cell more that once: when we move right or upward, the entry stays unchanged, when we move left or downward, the entry grows (by one at a step). To move from a cell back to the same cell, we have to make at least one move left of downward, and the entry cannot remain unchanged. Thus our path leads from our (arbitrarily chosen) cell to either a pillar or to a corner cell. Moving along this path in the opposite direction, we show that every condition $ C_{ij}$ follows either from one of the pillar conditions $ C_{i_1j_1},\dots,C_{i_Nj_N}$ or one of the conditions $ C_{0 0}, C_{nn}$, which are empty. Thus, the Schubert variety $ \X_{w}$ is determined by the conditions $ C_{i_1j_1},\dots,C_{i_Nj_N}$, that is, determined by the pillars.

To illustrate the above technique, let us calculate the tangent cone of the Schubert varieties corresponding to the Coxeter elements explicitly.

Proof.

We will need the following lemma We are grateful to M. Kashiwara for a simple proof.   × ¹.

Proof.

We are ready to prove Theorem 3.6, Part (ii). Assume that two permutations, $ w$ and $ w'$, are admissibly partially transpose to each other. We want to show that the tangent cones of $ \X_{w}$ and $ \X_{w'}$ coincide.

We can assume that $ w'=\trp_{t}(w)$ is an elementary partial transposition of $ w$, see Sect. 4.5 for the definition and the notation. The systems of equations for $ \X_{w}$ and $ \X_{w'}$ split in two parts: the equations coming from the pillar entries in the classes $ \Lc_{1}\sqcup \ldots \sqcup \Lc_{t}$ (these equations are a priori different for $ w$ and $ w'$ since the pillar entries are not in the same positions) and the equations coming from the pillar entries in the other classes, namely in $ \Lc_{t+1}\sqcup \ldots \sqcup \Lc_{s}$. The latter equations are identically the same for $ w$ and $ w'$.

Consider finally the two subsystems of equations for $ \X_{w}$ and $ \X_{w'}$ coming from the pillar entries in the set $ \Lc_{1}\sqcup \ldots \sqcup \Lc_{t}$. These two subsystems are precisely those describing the Schubert varieties associated to $ \tr_{t}(w)$ and $ \tr_{t}(w')$, respectively. These two varieties have same tangent cones since $ \tr_{t}(w)=\tr_{t}(w')^{-1}$. After intersecting with the tangent cone of the variety described by the rest of the system, one therefore obtains the same tangent cone for $ \X_{w}$ and $ \X_{w'}$.

Theorem 3.6 is proved.

In the case $ n=4$, the comparative number of Schubert varieties and their tangent cones, as a function of their dimension, is given by the following table. \begin{eqnarray*} \begin{array}{r|c|c|c|c|c|c|c} \dim&0&1&2&3&4&5&6\\ \hline \hbox{Schub}&1&3&5&6&5&3&1\\ \hline \hbox{TangCones}&1&3&3&3&3&2&1 \end{array} \end{eqnarray*} The total number of tangent cones in this case is $ 16$.

For $ n=5$, the table is: \begin{eqnarray*} \begin{array}{r|c|c|c|c|c|c|c|c|c|c|c} \dim&0&1&2&3&4&5&6&7&8&9&10\\ \hline \hbox{Schub}&1&4&9&15&20&22&20&15&9&4&1\\ \hline \hbox{TangCones}&1&4&6&7&9&9&10&8&6&2&1 \end{array} \end{eqnarray*} The total number of tangent cones is $ 63$.

For $ n=6$, the distribution of the tangent cones is as follows: \begin{eqnarray*} \begin{array}{r|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c} \dim&0&1&2&3&4&5&6&7&8&9&10&11&12&13&14&15\\ \hline \hbox{TangCones}&1&5&10&14&20&25&31&36&40&40&34&24&15&8&3&1 \end{array} \end{eqnarray*} The total number of tangent cones for $ n=6$ is $ 343$.

For $ n=7$ and $ 8$, the total numbers of tangent cones are: $ 1821$ and $ 13041$, respectively These numbers are obtained using computer programs.   × ². Note that the sequence $ 16,63,343,1821,13041,\ldots$ does not appear in Sloane's online Encyclopedia of Integer Sequences.

Let us also consider the case of small codimension.

The tangent cone of the Schubert variety $ \X_{w_0}$ corresponding to longest element $ w_0\in{}S_n$, is the only one tangent cone of dimension $ \frac{n(n-1)}{2}$.

Next, in the case of dimension $ \frac{n(n-1)}{2}-1$ ( i.e. , of codimension $ 1$), there are $ n-1$ Schubert varieties that have $ \left[\frac{n}{2}\right]$ tangent cones. Indeed, the elements $ \X_w$ and $ \X_{w^{-1}}$ have the same tangent cone.

There are $ \frac{(n+2)(n-1)}{2}$ Schubert varieties of codimension $ 2$. The number of their tangent cones depend on the parity of $ n$, as given by the following statement.

Proof.