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The cohomology rings of regular nilpotent Hessenberg varieties and Schubert polynomials.

1. Introduction. Let n be a positive integer. The (full) flag variety Fl([C.sup.n]) in [C.sup.n] is the collection of nested linear subspaces [V.sub.*] := ([V.sub.1] [subset] [V.sub.2] [subset] ... [subset] [V.sub.n] = [C.sup.n]) where each [V.sub.i] is an i-dimensional subspace in [C.sup.n]. We consider a weakly increasing function h : {1, 2, ..., n} [right arrow] {1, 2, ..., n} satisfying h(j) [greater than or equal to] j for j = 1, ..., n. This function is called a Hessenberg function. De Mari-ProcesiShayman ([6], [5]) defined a Hessenberg variety Hess(X, h) associated with a linear operator X : [C.sup.n] [right arrow] [C.sup.n] and a Hessenberg function h : {1, 2, ..., n} [right arrow] {1, 2, ..., n} as the following subvariety of the flag variety:

(1.1) Hess(X, h) := {[V.sub.*] [member of] Fl([C.sup.n]) | X[V.sub.i] [subset] [V.sub.h](i) for i = 1, 2, ..., n}.

We note that if h(j) = n for all j = 1, 2, ..., n or X is the zero matrix, then the corresponding Hessenberg variety coincides with the whole full flag variety Fl([C.sup.n]). The family of Hessenberg varieties also contains Springer varieties related to geometric representations of Weyl group ([19], [20]) and Peterson variety related to the quantum cohomology of the flag variety ([13], [16]). Recently, it has been found that Hessenberg varieties have surprising connection with other research areas such as hyperplane arrangements ([18], [2]) and graph theory ([17], [4], [9]).

In this paper we concentrate on Hessenberg varieties Hess(N, h) associated with a regular nilpotent operator N i.e. a matrix whose Jordan form consists of exactly one Jordan block with corresponding eigenvalue equal to 0. The Hessenberg variety Hess(N, h) is called a regular nilpotent Hessenberg variety. If we take h(j) = j + 1 for 1 [less than or equal to] j [less than or equal to] n-1 and h(n) = n, then the corresponding regular nilpotent Hessenberg variety is called the Peterson variety. Regular nilpotent Hessenberg varieties Hess(N, h) can be regarded as a (discrete) family of subvarieties of the flag variety connecting Peterson variety and the flag variety itself. The complex dimension of Hess(N, h) is [[summation].sup.n.sub.j=1] 1(h(j)-j) ([18]). A regular nilpotent Hessenberg variety is singular in general ([13], [12]). The cohomology ring of a regular nilpotent Hessenberg variety has been studied from various viewpoints (e.g. [3], [22], [11], [15], [7], [10], [1], [2]). To describe an explicit presentation of the cohomology ring of a regular nilpotent Hessenberg variety, polynomials [f.sub.i,j] were introduced in [1] as follows: (Throughout this paper we work with cohomology with coefficients in Q.) For 1 [less than or equal to] j [less than or equal to] i, we define a polynomial [f.sub.i,j] by

(1.2) [mathematical expression not reproducible].

Here, we take by convention [mathematical expression not reproducible]. whenever i = j. Then from the result of [1], the following isomorphism as Q-algebras holds

(1.3) H *(Hess(N, h)) [congruent to] Q[[x.sub.1], ..., [x.sub.n]]/([f.sub.h(j),j] | 1 [less than or equal to] j [less than or equal to] n).

Our main theorem is the following

Theorem 1.1. Let i, j be positive integers with 1 [less than or equal to] j < i [less than or equal to] n. Let [f.sub.i-1,j] be the polynomial in (1.2) and [G.sub.w] the Schubert polynomial for a permutation w in the symmetric group [S.sub.n]. Then we have

(1.4) [mathematical expression not reproducible]

where [w.sup.(i,j).sub.k] for 1 [less than or equal to] k [less than or equal to] i--j is a permutation in [S.sub.n] defined by

(1.5) [w.sup.(i,j).sub.k] := ([s.sub.i-k] [s.sub.i-k-i] ... [s.sub.j]) ([s.sub.i-k+1] [s.sub.i-k+2] ... [s.sub.i-l]).

Here, [s.sub.r] denotes the transposition of r and r + 1 for r = 1, 2, ..., n - 1 and we take by convention ([s.sub.i-k+1] [s.sub.i-k+2] ... [s.sub.i-1]) = id whenever k = 1.

We can interpret the equality (1.4) in Theorem 1.1 from a geometric viewpoint under the circumstances of having a codimension one Hessenberg variety Hess(N, h') in the original Hessenberg variety Hess(N, h). We will discuss more details in Section 4.

2. Divided difference operator. In this section, we observe a new property of polynomials [f.sub.i,j] in (1.2) related with the divided difference operator defined by Bernstein-Gelfand-Gelfand and Demazure. This is the key property for the proof of the main theorem. We first recall the definition of the divided difference operator and the Schubert polynomials. For general reference, see [8].

Let f be a polynomial in Z[[x.sub.1], ..., [x.sub.n]] and [s.sub.i] the transposition of i and i + 1 for any i = 1, 2, ..., n - 1. Let [s.sub.i](f) denote the result of interchanging [x.sub.i] and [x.sub.i+1] in f. Then the divided difference operator [[partial derivative].sub.i] on the polynomial ring Z[[x.sub.1], ..., [x.sub.n]] is defined by the formula

(2.1) [[partial derivative].sub.i](f) := f - [s.sub.i]/[x.sub.i] - [x.sub.i+1].

Since f - [s.sub.i](f) is divisible by [x.sub.i] - [x.sub.i+1], [[partial derivative].sub.i](f) is always a polynomial. If f is homogeneous of degree d, then [[partial derivative].sub.i](f) is homogeneous of degree d - 1.

For a reduced expression [mathematical expression not reproducible], we set [mathematical expression not reproducible]. Since the divided difference operators satisfy the relations [[partial derivative].sub.i] [[partial derivative].sub.i+1] [[partial derivative].sub.i] = [[partial derivative].sub.i+1] [[partial derivative].sub.i] [[partial derivative].sub.i+1] and [[partial derivative].sup.2.sub.i] = 0, the operator [[partial derivative].sub.u] is independent of the choice of reduced expressions for u. The Schubert polynomial [G.sub.w] for a permutation w in the symmetric group [S.sub.n] is defined as follows. For [w.sub.0] = [n, n - 1, ..., 1] [member of] [S.sub.n] the permutation of the longest length in one-line notation, we define

[mathematical expression not reproducible].

For general permutation w in [S.sub.n], write [mathematical expression not reproducible] with l([mathematical expression not reproducible]) = l([w.sub.0])- p for 1 [less than or equal to] p [less than or equal to] r. Then the Schubert polynomial is inductively defined by

[mathematical expression not reproducible].

In general, Schubert polynomials have the following property

(2.2) [mathematical expression not reproducible].

Note that the Schubert polynomial [G.sub.w] is a homogeneous polynomial in Z[[x.sub.1], ..., [x.sub.n-1]] of degree l(w) which is the number of inversions in w, called the length of w, i.e.,

l(w) = #{j < i < | w(j) > w(i)}.

Schubert polynomials have an important property that [G.sub.w] is in fact independent of n in the following sense. For w [member of] [S.sub.n] and m [greater than or equal to] n, we define [w.sup.(m)] [member of] [S.sub.m] by [w.sup.(m)] (i) = w(i) for 1 [less than or equal to] i [less than or equal to] n and [w.sup.(m)] (i) = i for n + 1 [less than or equal to] i [less than or equal to] m. Then we have

(2.3) [mathematical expression not reproducible].

The following proposition is the key property for the proof of the main theorem.

Proposition 2.1. Let i, j be positive integers with j < i. Let [f.sub.i,j] be the polynomial in (1.2) and [[partial derivative].sub.i] the divided difference operator in (2.1). Then

(2.4) [[partial derivative].sub.j] ([f.sub.i,j]) = [f.sub.i,j+1],

(2.5) [[partial derivative].sub.i] ([f.sub.i,j]) = -[f.sub.i-1,j].

Proof. We first prove the equality (2.4). Since

[mathematical expression not reproducible],

the difference [f.sub.i,j] - [s.sub.j]([f.sub.i, j]) is

([mathematical expression not reproducible].

Hence, we obtain [[partial derivative].sub.j] ([f.sub.i,j]) = [f.sub.i,j+1].

We next prove the equality (2.5). Since

[mathematical expression not reproducible],

we have

[mathematical expression not reproducible].

Thus, we obtain

[mathematical expression not reproducible].

From Proposition 2.1, we see that every polynomial [f.sub.i,j] for 1 [less than or equal to] j [less than or equal to] i [less than or equal to] n is obtained from the single polynomial [f.sub.n,1] by using the divided difference operator. More concretely, we set

[F.sub.n] := [f.sub.n,1] = ([x.sub.1] - [x.sub.n])([x.sub.1] - [x.sub.n-1]) ... ([x.sub.1] - [x.sub.2])[x.sub.1].

Then we obtain

[mathematical expression not reproducible].

3. Proof of Theorem 1.1. In this section we prove Theorem 1.1. To do that, we need Monk's formula.

Theorem 3.1 (Monk's formula [14], see also [8] p. 180). Let [G.sub.w] be the Schubert polynomial for w [member of] [S.sub.n] and [s.sub.r] the transposition of r and r + 1. Then we have

(3.1) [mathematical expression not reproducible]

where [t.sub.pq] is the transposition interchanging values of p and q, and the sum is over all 1 [less than or equal to] p [less than or equal to] r < q such that w(p) < w(q) and w(i) is not in the interval (w(p), w(q)) for any i in the interval (p, q).

Using Monk's formula, we first prove the following proposition which is the case i = n and j = 1 of Theorem 1.1.

Proposition 3.2. Let n > 1 and [f.sub.n-1,1] the polynomial in (1.2). Let [G.sub.w] be the Schubert polynomial for w [square root of [member of] [S.sub.n]. Then we have

[mathematical expression not reproducible]

where [w.sup.n-1.sub.k] [member of] [S.sub.n] is the permutation defined in (1.5).

Proof. We prove the proposition by induction on n. For the base case n = 2, it holds because [mathematical expression not reproducible]. Now we assume n > 2 and the following equality

[mathematical expression not reproducible]

Since we have from the definition (1.2) that

[mathematical expression not reproducible],

it is enough to prove the following equality

(3.2) [mathematical expression not reproducible].

We prove the equality (3.2) using Monk's formula (3.1).

Case(i): Using Monk's formula (3.1), the product [mathematical expression not reproducible] equals

[mathematical expression not reproducible],

Case(ii): Using Monk's formula (3.1), the product [mathematical expression not reproducible] equals

[mathematical expression not reproducible].

Case(iii): Using Monk's formula (3.1), the product [mathematical expression not reproducible] equals

[mathematical expression not reproducible].

From Case(i), (ii), (iii) together with equalities [w.sup.(n-1, 1).sub.1] [t.sub.1n] = [w.sup.(n, 1).sub.k] and [w.sup.(n-1, 1).sub.k] [t.sub.n-1n] = [w.sub.(n, 1).sub.k+1] for 1 [less than or equal to] k [less than or equal to] n - 2, the left hand side of (3.2) reduces to

[mathematical expression not reproducible].

However, since we have [w.sup.(n-1,1).sub.k+1] [t.sub.1n-k-1] = [w.sup.(n-1,1).sub.k] [t.sub.n-k-1n-1] for 1 [less than or equal to] k [less than or equal to] n - 3, the above expression is equal to the right hand side of (3.2). Therefore, we obtain (3.2). This completes the induction step and proves the proposition. ?

Proof of Theorem 1.1. We now prove Theorem 1.1 by induction on j. For the base case j = 1, it holds from Proposition 3.2 together with the property (2.3) of Schubert polynomials. Now we assume j > 1 and the following equality

[mathematical expression not reproducible].

Then since

[w.sup.(i,j-1).sub.k] (j - 1) > [w.sup.(i,j-1)] (j) if 1 [less than or equal to] k [less than or equal to] i - j, [w.sup.(i,j-1).sub.k] (j - 1) < [w.sup.(i,j-1)] (j) if k = i - j + 1,

we have from (2.2) that

[mathematical expression not reproducible].

Therefore, using (2.4), we obtain

[mathematical expression not reproducible].

This completes the induction step and proves Theorem 1.1.

4. Geometric meaning of Theorem 1.1.

In this section we observe a geometric meaning of Theorem 1.1. Throughout this section we use the notation

[n] :={1, 2, ..., n}.

Recall that a Hessenberg function h : [n] [right arrow] [n] is a weakly increasing function satisfying h(j) [greater than or equal to] j for j [member of] [n]. We denote a Hessenberg function h by listing its values in sequence, i.e.

h = (h(1), h(2), ..., h(n)).

We often regard a Hessenberg function as a configuration of boxes on a square grid of size n x n whose shaded boxes correspond to the boxes in the position (i, j) for i, j [member of] [n] and i [less than or equal to] h(j).

Example 4.1. Let n = 5. A function h = (3, 3, 4, 5, 5) is a Hessenberg function and the corresponding configuration of boxes on a square grid of size 5 x 5 is given in Figure 1.

An (i j)-th box of a Hessenberg function is a corner if there is neither a shaded box in (i + 1, j) nor in (i, j - 1). Let h : [n] [right arrow] [n] be a Hessenberg function with (i, j)-th box as a corner with i > j. We define a Hessenberg function h' : [n] [right arrow] [n] by removing (i, j)-th box of h (see Figure 2). More precisely, h' is defined by

h'(k) = h(k) if k [not equal to] j, h'(j) = h(j) - 1 = i - 1.

Then, we have Hess (N, h') [subset] Hess (N, h) by the definition (1.1). From the isomorphism (1.3) we obtain

[f.sub.i-1,j] [not equal to] 0 in [H.sup.*] (Hess(N, h)) , [f.sub.i-1,j] = 0 in [H.sup.*] (Hess(N, h')).

In fact, suppose for a contradiction that [f.sub.i-1, j] = 0 in [H.sup.*] (Hess(N, h)). Then the ideal ([f.sub.h(1), 1], ..., [f.sub.h(n),n]) is equal to the ideal ([f.sub.h'](1), 1, ..., [f.sub.h'] (n) ,n). It follows from (1.3) that [H.sup.*] (Hess(N, h)) is isomorphic to [H.sup.*] (Hess(N, h')). This contradicts the equality dim Hess (N, h) = [[summation].sup.n.sub.j=1] (h(j) - j) = dimHess(N, h') + 1.

Next, we consider intersections of a regular nilpotent Hessenberg variety and Schubert cells. We first recall the definition of Schubert cells. Let G be the general linear group GL(n, C) and B the standard Borel subgroup of upper-triangular invertible matrices. Then the flag variety Fl([C.sup.n]) can be realized as a homogeneous space G/B. For a permutation w in the symmetric group [S.sub.n], we define the Schubert cell [X.sup.[omicron].sub.w] of the flag variety by [X.sup.[omicron].sub.w] = BwB/B. The Schubert cell [X.sup.[omicron].sub.w] is isomorphic to an affine space [C.sup.l(w)]. It follows from [21, Theorem 6.1] that the condition for Hess(N, h) [intersection] [X.sup.[omicron].sub.w] being nonempty is given by

(4.1) Hess(N, h) [intersection] [X.sup.[omicron].sub.w] [not equal to] 0 [??] [w.sup.-1] (w(r) - 1) [less than or equal to] h(r) for all r [member of] [n].

The following lemma gives the geometric meaning of the permutations [w.sup.(i,j).sub.k] in (1.5).

Lemma 4.2. Let h : [n] [right arrow] [n] be a Hessenberg function with (i, j)-th box as a corner with i > j and h' : [n] [right arrow] [n] a Hessenberg function obtained from h by removing (i, j)-th box. Let {[X.sup.[omicron].sub.w]} be Schubert cells. Then, the set of permutations [w.sup.(i, j).sub.k] (1 [less than or equal to] k [less than or equal to] i - j) in (1.5) coincides with the set of minimal length permutations w in Sn such that

Hess(N, h) [intersection] [X.sup.[omicron].sub.w] [not equal to] 0 and Hess(N, h') [intersection] [X.sup.[omicron].sub.w] = 0.

Proof. Let [X.sup.[omicron].sub.w] be a Schubert cell. It follows from (4.1) that a necessary and sufficient condition for Hess(N, h) [intersection] [X.sup.[omicron].sub.w] [not equal to] 0 and Hess(N, h') [intersection] [X.sup.[omicron].sub.w] = 0 is given by i - 1 = h'(j) < [w.sup.-1] (w(j) - 1) [less than or equal to] h(j) = i and [w.sup.-1] (w(r) - 1) [less than or equal to] h(r) for r [not equal to] j, that is,

(4.2) w(j) -1 = w(i) ,

(4.3) [w.sup.-1] (w(r) - 1) [less than or equal to] h(r) for r [not equal to] j.

It is clear that [w.sup.(i, j).sub.k] satisfies (4.2) and (4.3). Let v be a permutation in [S.sub.n] satisfying (4.2) and (4.3) with minimal length, and we prove that v is a permutation [w.sup.(i, j).sub.k] for some 1 [less than or equal to] k [less than or equal to] i - j. From the minimality of the number of inversions of v, we must arrange the values v(r) for r [not equal to] j, i in one-line notation as a subsequence in the increasing order. If v(j) = m + 1 , v(i) = m for some m with 1 [less than or equal to] m [less than or equal to] j - 1 or i [less than or equal to] m [less than or equal to] n - 1, then l(v) > l([w.sup.(i, j).sub.k]) = i - j. This contradicts the minimality for the length of v. Hence, we have v(j) = i - k + 1 , v(i) = i - k for some k with 1 [less than or equal to] k [less than or equal to] i - j. This means that v = [w.sup.(i, j).sub.k].

In summary, we can observe a geometric meaning of Theorem 1.1 as follows. Let Hess(N , h) be a regular nilpotent Hessenberg variety. By removing an (i j)-th box from h, we obtain the new Hessenberg variety Hess(N, h'). Then, Lemma 4.2 tells us that [mathematical expression not reproducible] are the minimal dimensional Sckhubert cells which do not intersect with the new Hessenberg variety Hess(N, h'). Theorem 1.1 now says that an alternating sum of Schubert classes [mathematical expression not reproducible] vanishes in [H.sup.*] (Hess(N, h')) as the new relation which we do not have in [H.sup.*] (Hess(N, h)).

doi: 10.3792/pjaa.94.87

Acknowledgements. The author is grateful to Hiraku Abe for fruitful discussions and comments on this paper. The author learned the equality (2.5) from him. The author also appreciates Mikiya Masuda for his support and valuable comments on this paper. The author is partially supported by JSPS Grant-in-Aid for JSPS Fellows: 17J04330.

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By Tatsuya HORIGUCHI

Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, 1-5 Yamadaoka, Suita, Osaka 565-0871, Japan

(Communicated by Masaki KASHIWARA, M.J.A., Oct. 12, 2018)

2010 Mathematics Subject Classification. Primary 14N15.

Caption: Fig. 1. The configuration of shaded boxes for h = (3, 3, 4, 5, 5).

Caption: Fig. 2. The pictures of h and h'.
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