THE S 1 -EQUIVARIANT COHOMOLOGY RINGS OF (n k, k) SPRINGER VARIETIES

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1 Horiguchi, T. Osaka J. Math. 52 (2015), THE S 1 -EQUIVARIANT COHOMOLOGY RINGS OF (n k, k) SPRINGER VARIETIES TATSUYA HORIGUCHI (Received January 6, 2014, revised July 14, 2014) Abstract The main result of this note gives an explicit presentation of the S 1 -equivariant cohomology ring of the (n k, k) Springer variety (in type A) as a quotient of a polynomial ring by an ideal I, in the spirit of the well-known Borel presentation of the cohomology of the flag variety. Contents 1. Introduction Nilpotent Springer varieties and S 1 -fixed points Main theorem Proof of the main theorem References Introduction The Springer variety S N associated to a nilpotent operator NÏ n n is the subvariety of Flags( n ) defined as S N {V ¾ Flags( n ) N V i V i 1 for all 1 i n} where V denotes a nested sequence 0 V 0 V 1 V n 1 V n n of subspaces of n and dim V i i for all i. When N consists of two Jordan blocks of sizes n k and k with n 2k, we denote S N by S (n k,k). The cohomology ring of Springer variety S N has been much studied due to its relation to representations of the permutation group on n letters ([5], [6]). In fact, the ordinary cohomology ring H (S N ÁÉ) is known to be the quotient of a polynomial ring by an ideal called Tanisaki s ideal ([7]). In this paper we study the equivariant cohomology ring of S (n k,k) with respect to a certain circle action on S N which we describe below. Recall that the n-dimensional compact torus T consisting of diagonal unitary matrices of size n acts on Flags( n ) in a natural way. A certain circle subgroup S of T 2010 Mathematics Subject Classification. Primary 55N91; Secondary 05A17.

2 1052 T. HORIGUCHI leaves S N invariant (cf. Section 2). The ring homomorphism H T (Flags(n )Á É) H S (S NÁ É) induced from the inclusions of S N into Flags( n ) and S into T is known to be surjective (cf. [3]). The main result of this paper is an explicit presentation of H S (S (n k,k)áé) as a ring using the epimorphism above (Theorem 3.3). In related work, Dewitt and Harada [1] give a module basis of H S (S (n k,k)á É) over H (BSÁ É) when k 2 from the viewpoint of Schubert calculus. Finally, since the restriction map H S (S NÁ É) H (S N Á É) is also known to be surjective for any nilpotent operator N, our presentation of H S (S (n k,k)á É) yields a presentation of H (S (n k,k) Á É) as a ring (Corollary 3.4). However, the resulting presentation is slightly different from the one given in [7]. This paper is organized as follows. We briefly recall the necessary background in Section 2. Our main theorem, Theorem 3.3, is formulated in Section 3 and proved in Section Nilpotent Springer varieties and S 1 -fixed points We begin by recalling the definition of the nilpotent Springer varieties in type A. Since we work exclusively with type A in this paper, we henceforth omit it from our terminology. The flag variety Flags( n ) is the projective variety of nested subspaces in n, i.e. Flags( n ) {V (0 V 0 V 1 V n 1 V n n ) dim V i i}. Let NÏ n n be a nilpotent operator. The (nilpotent) Springer DEFINITION. variety S N associated to N is defined as S N {V ¾ Flags( n ) N V i V i 1 for all 1 i n}. Since S gng 1 is homeomorphic (in fact, isomorphic as algebraic varieties) to S N for any g ¾ GL n (), we may assume that N is in Jordan canonical form with Jordan blocks of weakly decreasing sizes. Let N denote the partition of n with entries the sizes of the Jordan blocks of N. The n-dimensional torus T consisting of diagonal unitary matrices of size n acts on Flags( n ) in a natural way and the circle subgroup

3 COHOMOLOGY RINGS OF (n k, k) SPRINGER VARIETIES 1053 S of T defined as (2.1) S ¼ g g 2... g n ½ g ¾, g 1 leaves S N Flags( n ) invariant (see [2]). The T -fixed point set Flags( n ) T of Flags( n ) is given by {(e Û(1) e Û(1), e Û(2) e Û(1), e Û(2),, e Û(n) n ) Û ¾ S n } where e 1, e 2,, e n is the standard basis of n and S n is the permutation group on n letters {1, 2,, n}, so we identify Flags( n ) T with S n as is standard. Also, since the S-fixed point set Flags( n ) S of Flags( n ) agrees with Flags( n ) T, we have S S N S N Flags( n ) S S N Flags( n ) T S n. We denote by S (n k,k) the Springer variety corresponding to the partition N (n k, k) with 2k n. We next describe the S-fixed points in S (n k,k). Let Û l1,l 2,,l k be an element of S n defined by (2.2) Û l1,l 2,,l k (i) n k j if i l j, i j if l j i l j1, where l 0 Ï 0, l k1 Ï n 1. Note that Û 1 l 1,l 2,,l k (i) Û 1 l 1,l 2,,l k (i ¼ ) if 1 i i ¼ n k or n k 1 i i ¼ n. Take n 4 and k 2. Using one-line notation, the set of permuta- EXAMPLE. tions of the form described in (2.2) are as follows: [3, 4, 1, 2], [3, 1, 4, 2], [3, 1, 2, 4], [1, 3, 4, 2], [1, 3, 2, 4], [1, 2, 3, 4]. Lemma 2.1. The S-fixed points S S (n k,k) of the Springer variety S (n k,k) is the set Proof. {Û l1,l 2,,l k ¾ S n 1 l 1 l 2 l k n}. Since S S (n k,k) Flags( n ) T, any element V of S S (n k,k) is of the form V (e Û(1) e Û(1), e Û(2) e Û(1), e Û(2),, e Û(n) ) for some Û ¾ S n. Since N is the nilpotent operator consisting of two Jordan blocks with weakly decreasing sizes (n k, k), Ne i 0 if i 1 or n k 1, e i 1 otherwise.

4 T. HORIGUCHI Therefore, if V belongs to S (n k,k), then Û(1) 1 or n k 1. If Û(1) 1 then Û(2) 2 or n k 1. If Û(1) n k 1 then Û(2) 1 or n k 2, and so on. for some 1 l 1 l 2 l k n. Conversely, one This shows that Û Û l1,l 2,,l k can easily see that Û l1,l 2,,l k ¾ S(n S k,k). 3. Main theorem In this section, we formulate our main theorem which gives an explicit presentation of the S-equivariant cohomology ring of the (n k, k) Springer variety. First, we recall an explicit presentation of the T -equivariant cohomology ring of the flag variety. Let E i be the subbundle of the trivial vector bundle Flags( n ) n over Flags( n ) whose fiber at a flag V is just V i. We denote the T -equivariant first Chern class of the line bundle E i E i 1 by Æx i ¾ H 2 T (Flags(n )Á É). The torus T consisting of diagonal unitary matrices of size n has a natural product decomposition T (S 1 ) n where S 1 is the unit circle of. This decomposition identifies BT with (BS 1 ) n and induces an identification H T (ptá É) H (BTÁ É) Ç H (BS 1 Á É) É[t 1,, t n ], where t i (1 i n) denotes the element corresponding to a fixed generator t of H 2 (BS 1 ÁÉ). Then H T (Flags(n )ÁÉ) is generated by Æx 1,,Æx n,t 1,,t n as a ring. We define a ring homomorphism from the polynomial ring É[x 1,, x n ] to H T )Á (Flags(n É) by (x i ) Æx i. It is known that is an epimorphism and Ker is generated as an ideal by e i (x 1,, x n ) e i (t 1,, t n ) for all 1 i n, where e i is the ith elementary symmetric polynomial. Thus, we have an isomorphism: (3.1) H T (Flags(n )Á É) É[x 1,, x n, t 1,, t n ](e i (x 1,, x n ) e i (t 1,, t n ), 1 i n). We consider the following commutative diagram: H T (Flags(n )Á É) H T (Flags(n ) T Á É) Ä Û¾S n É[t 1,, t n ] 1 H S (S NÁ É) 1 2 H S (S S NÁ É) Ä Û¾S S N S n É[t] where all the maps are induced from inclusion maps, and we have an identification H S (ptá É) H (BSÁ É) H (BS 1 Á É) É[t] where we identify S with S 1 through the map diag(g, g 2,, g n ) g. The maps 1 and 2 in (3.1) are injective since the odd degree cohomology groups of Flags( n ) and

5 COHOMOLOGY RINGS OF (n k, k) SPRINGER VARIETIES 1055 S N vanish. The map 1 in (3.1) is known to be surjective (cf. [3]) and the map 2 is obviously surjective. Since 1 is surjective, we have the following lemma. Let i be the image 1 (Æx i ) of Æx i for each i. Lemma 3.1. The S-equivariant cohomology ring H S (S NÁÉ) is generated by 1,, n, t as a ring where i is the image of Æx i under the map 1 in (3.1). We next consider relations between 1,, n, and t. We have 2 ( i ) Û Û(i)t because 1 (Æx i ) Û t Û(i), 1 (t i ) Û t i, and 2 (t i ) it, where f Û denotes the Û-component of f ¾ Ä Û¾S n É[t 1,, t n ]. (3.2) (3.3) (3.4) Lemma 3.2. where in The elements 1,, n, t satisfy the following relations: i n(n 1) t 0, 2 ( i i 1 (n k i)t)( i i 1 t) 0 (1 i n), 0 jk ( i j (i j j)t)) 0 (1 i 0 i k n), Proof. The relation (3.2) follows from a relation in H T (Flags(n )Á É). In fact, 1in i n(n 1) t 1 ((e 1 (Æx 1,, Æx n ) e 1 (t 1,, t n ))) 0. 2 In the following, we denote 2 ( i ) by the same notation i the relation (3.3), it is sufficient to prove either for each i. To prove (3.5) ( i i 1 (n k i)t) Ûl1,l 2,,l k 0 or ( i i 1 t) Ûl1,l 2,,l k 0 for any Û l1,l 2,,l k ¾ S(n S k,k) since the restriction map 2 in (3.1) is injective. We first treat the case i 1. By the definition of Û l1,l 2,,l k in (2.2) the following holds: 1 Ûl1,l 2,,l k Û l1,l 2,,l k (1)t (n k 1)t if l 1 1, t if l 1 1. This shows (3.5) for i 1 because 0 0.

6 1056 T. HORIGUCHI (3.6) (3.7) We now treat the case 1 i n. Note that ( i i 1 ) Ûl1,l 2,,l k (Û l1,l 2,,l k (i) Û l1,l 2,,l k (i 1))t, ( i i 1 ) Ûl1,l 2,,l k (Û l1,l 2,,l k (i)û l1,l 2,,l k (i 1))t. We take four cases depending on whether i 1 and i appear in l 1,, l k or not. (i) If l j i 1 i l j1 for some 1 j k 1, then by (2.2) and (3.6), ( i i 1 ) Ûl1,l 2,,l k ((n k j 1) (n k j))t t. (ii) If l j i 1 i l j1 for some 0 j k, then by (2.2) and (3.6), ( i i 1 ) Ûl1,l 2,,l k ((i j) (i j 1))t t. (iii) If l j i 1 i l j1 for some 1 j k, then by (2.2) and (3.7), ( i i 1 ) Ûl1,l 2,,l k ((i j)(n k j))t (n k i)t. (iv) If l j 1 i 1 i l j for some 1 j k, then by (2.2) and (3.7), ( i i 1 ) Ûl1,l 2,,l k ((n k j)(i j))t (n k i)t. Therefore, (3.5) holds in all cases, proving the relations (3.3). Finally we prove the relations (3.4). For any Û l1,l 2,,l k ¾S S (n k,k) integer i j such that l j i j l j1 for some 0 j k. Thus, we have Û l1,l 2,,l k (i j ) i j j., there is a positive This means that 0 jk ( i j (i j j)t) Ûl1,l 2,,l k 0. Therefore, the relations (3.4) hold, and the proof is complete. It follows from Lemma 3.2 that we obtain a well-defined ring homomorphism (3.8) ³Ï É[x 1,, x n, t]i H S (S (n k,k)á É) where I is the ideal of a polynomial ring É[x 1,, x n, t] generated by the following three types of elements: (3.9) (3.10) (3.11) 1in x i n(n 1) t, 2 (x i x i 1 (n k i)t)(x i x i 1 t) (1 i n), 0 jk (x i j (i j j)t) (1 i 0 i k n),

7 COHOMOLOGY RINGS OF (n k, k) SPRINGER VARIETIES 1057 where x 0 0. Moreover, ³ is surjective by Lemma 3.1. The following is our main theorem and will be proved in the next section. Theorem 3.3. Let S (n k,k) be the (n k, k) Springer variety with 0k n2 and let the circle group S act on S (n k,k) as described in Section 2. Then the S-equivariant cohomology ring of S (n k,k) is given by H S (S (n k,k)á É) É[x 1,, x n, t]i where H S (ptá É) É[t] and I is the ideal of the polynomial ring É[x 1,, x n, t] generated by the elements listed in (3.9), (3.10), and (3.11). Since the ordinary cohomology ring of S (n k,k) can be obtained by taking t 0 in Theorem 3.3, we obtain the following corollary. Corollary 3.4. the ordinary cohomology ring of S (n k,k) is given by Let S (n k,k) be (n k, k) Springer variety with 0 k n2. Then H (S (n k,k) Á É) É[x 1,, x n ]J where J is the ideal of the polynomial ring É[x 1,, x n ] generated by the following three types of elements: 1in x 2 i x i, 1 jk1 (1 i n), x i j (1 i 1 i k1 n). REMARK. A ring presentation of the cohomology ring of the Springer variety S N is given in [7] for an arbitrary nilpotent operator N. Specifically, it is the quotient of a polynomial ring by an ideal called Tanisaki s ideal. When N (n k, k), Tanisaki s ideal is generated by the following three types of elements: e 1 (x 1,, x n ), e 2 (x i1,, x in 1 ) e k1 (x i1,, x ik1 ) (1 i 1 i n 1 n), (1 i 1 i k1 n), where e i is the ith elementary symmetric polynomial. Note that the first and third elements above are the same as those in Corollary 3.4. In fact, one can easily check that Tanisaki s ideal above agrees with the ideal J in Corollary 3.4 although the generators are slightly different.

8 1058 T. HORIGUCHI 4. Proof of the main theorem This section is devoted to the proof of Theorem 3.3. More precisely, we will prove that the epimorphism ³ in (3.8) is an isomorphism. For this, we first find generators of É[x 1,, x n, t]i as a É[t]-module. Recall that a filling of by the alphabet {1,, n} is an injective placing of the integers {1,, n} into the boxes of. DEFINITION. Let be a Young diagram with n boxes. A filling of is a permissible filling if for every horizontal adjacency a b we have a b. Also, a permissible filling is a standard tableau if for every vertical adjacency a b we have a b. Let T be a permissible filling of (n l, l) with 0 l k. Let j 1, j 2,, j l be the numbers in the bottom row of T. We define x T Ï x j1 x j2 x jl and x T0 Ï 1 where T 0 is the standard tableau on (n). Proposition 4.1. The set {x T T standard tableau on (n l, l) with 0 l k} generates É[x 1,, x n, t]i as a É[t]-module. Proof. It is sufficient to prove that x b1 x b2 x bl (1 b 1 b 2 b l n) can be written in É[x 1,, x n, t]i as a É[t]-linear combination of the x T where T is a standard tableau. We prove this by induction on l. The base case l 0 is clear. Now we assume that l 1 and the claim holds for l 1. The relations (3.10) imply that (4.1) x 2 i (n k i 1)tx i t 1pi 1 x p 1pi (n k p)t 2 (1 i n) by an inductive argument on i, so we may assume b 1 b 2 b l. To prove the claim for l, we consider two cases: 1 l k and l k 1. CASE (i) Suppose 1 l k. We write x b1 x b2 x bl x U where U a 1 a l a l1 a n l b 1 b l is a permissible filling of (n l, l). Let j be the minimal positive integer in the set {r a r b r, 1 r l}, i.e., (4.2) (4.3) a i b i a j b j. (1 i j),

9 COHOMOLOGY RINGS OF (n k, k) SPRINGER VARIETIES 1059 We consider the following equation which follows from the relation (3.9): (4.4) ( x a1 x a2 x a j 1 ) j x b j1 x bl x b1 x b2 x bl x a j x a j1 x an l n(n 1) j t 2 xb j1 x. bl Claim 1. The left hand side in (4.4) is a É[t]-linear combination of the x T where the T are standard tableaux. Proof. We expand the left hand side in (4.4). Then any monomial which appears in the expansion is of the form x «1 a 1 x «j 1 a j 1 x b j1 x bl where È j 1 i1 «i j and «i 0. Note that «i 1 for some i since È j 1 i1 «i j and «i 0. Therefore, using the relations (4.1), the monomial above turns into a sum of elements of the form f (t) x c1 x ch where h l, 1 c 1 c h n, and f (t) ¾ É[t], and by the induction assumption the term above can be written as a É[t]-linear combination of the x T where T is a standard tableau. This proves Claim 1. Claim 2. The right hand side in (4.4) can be written as a É[t]-linear combination of x U and monomials x T and x U ¼ where the coefficient of x U is equal to 1, T is a standard tableau on shape (n l, l) and U ¼ is a permissible filling of (n l, l) such that each of the leftmost j columns are strictly increasing (i.e. a r b r, 1 r j). Proof. We expand the right hand side in (4.4). A monomial which appears in this expansion is of the form x 1 b p1 x m b pm x «1 a q1 x «h a qh x b j1 x bl È m where i1 i È h i1 «i j, i 1, «i 1 and 1 p 1 p m l, j q 1 q h È m n l. It is enough to consider the case i1 «i j since if È m i1 i È h i1 «i j then it follows from the induction assumption that the above form can be written as a É[t]-linear combination of the x T where T is a standard i1 i È h tableau. If p m j 1 or some i or «i is more than 1, then it follows from the relations (4.1) and the induction assumption that the monomial above can be written as a linear combination of x T s over É[t] where T is a standard tableau. If p m j and all i and «i are equal to 1, then h j m and the monomial above is of the form x bp1 x bpm x aq1 x aq j m x b j1 x bl

10 1060 T. HORIGUCHI where 1 p 1 p m j q 1 q j m n l. This monomial is associated to a permissible filling U ¼ given by where and U ¼ c 1 c l c l1 c n l d 1 d l d i b pi if 1 i m, min{{a q1,, a q j m, b j1,, b l } {d m1,, d i 1 }} if m i l, c i min{{a 1,, a n l, b 1,, b j } {a q1,, a q j m, b p1,, b pm, c 1,, c i 1 }} for 1 i n l. Note that x U ¼ x U if and only if m j, since m j d i b i for 1 i l. We consider the case m j. Since j q 1 and a j b j by (4.3), we have c i min{{a 1,, a j 1, b 1,, b j } {b p1,, b pm, c 1,, c i 1 }} for 1 i j. If 1 i m, we have c i a i b i b pi d i. If m i j, we have c i max{a j 1, b j } min{a j, b j1 } d i by (4.2), (4.3), and j q 1. Thus, U ¼ is a permissible filling of (n l, l) such that each of the leftmost j columns are strictly increasing (i.e. a r b r, 1 r j). This proves Claim 2. Claims 1 and 2 show that x U can be written as a É[t]-linear combination of x U ¼ and x T, where U ¼ and T are as above. Applying the above discussion for x U ¼ in place of x U, we see that x U ¼ can be written as a É[t]-linear combination of x U ¼¼ and x T where U ¼¼ is a permissible filling of (n l, l) such that each of the leftmost j 1 columns are strictly increasing (i.e. a r b r, 1 r j 1) and T is a standard tableau. Repeating this procedure, we can finally express x U as a É[t]-linear combination of the x T where T is a standard tableau. CASE (ii) If l k 1, it follows from the relations (3.11) and the induction assumption that x b1 x b2 x bl can be expressed as a É[t]-linear combination of the x T where T is a standard tableau. This completes the induction step and proves the proposition. Recall that for a box b in the ith row and jth column of a Young diagram, h(i, j) denote the number of boxes in the hook formed by the boxes below b in the jth column, the boxes to the right of b in the ith row, and b itself. EXAMPLE. For the Young diagram and the box in the (2, 1) location,

11 COHOMOLOGY RINGS OF (n k, k) SPRINGER VARIETIES 1061 the hook is and h(2, 1) 6. Lemma 4.2. Let be a Young diagram. Let f denote the number of standard tableaux on. Then n k 0lk f (n l,l). Proof. We prove the lemma by induction on k. As the case k 0 is clear, we assume that k 1 and that the lemma holds for k 1. We use the following hook length formula: f n! (i, j)¾ h(i, j). Using the induction assumption and the hook length formula, we have 0lk f (n l,l) 0lk 1 n n. k k 1 f (n l,l) (n k,k) f n! (n 2k 1) (n k 1)! k! This completes the induction step and proves the lemma. It follows from Proposition 4.1 and Lemma 4.2 that rank É[t] É[x 1,, x n, t]i 0lk f (n l,l) n k On the other hand, since the odd degree cohomology groups of S N. vanish, we have an isomorphism H S (S NÁ É) É[t]Å H (S N Á É) as É[t]-modules, and the cellular decomposition of S N given by Spaltenstein [4] (cf. also Hotta Springer [3]) implies that dim H (S N Á É) n N Ï n 1! 2! r! where N ( 1, 2,, r ). These show rank É[t] H S (S (n k,k)á É) dim É H (S (n k,k) Á É) n k.

12 1062 T. HORIGUCHI Therefore, we have rank É[t] É[x 1,, x n, t]i rank É[t] H S (S (n k,k)á É). This means that the epimorphism ³ in (3.8) is actually an isomorphism, proving Theorem 3.3. ACKNOWLEDGEMENTS. The author thanks Professor Mikiya Masuda for valuable discussions and useful suggestions, and thanks Professor Megumi Harada for useful suggestions, and thanks Yukiko Fukukawa for valuable discussions. References [1] B. Dewitt and M. Harada: Poset pinball, highest forms, and (n 2, 2) Springer varieties, Electron. J. Combin. 19 (2012), paper 56. [2] M. Harada and J. Tymoczko: Poset pinball, GKM-compatible subspaces, and Hessenberg varieties, arxiv: [3] R. Hotta and T.A. Springer: A specialization theorem for certain Weyl group representations and an application to the Green polynomials of unitary groups, Invent. Math. 41 (1977), [4] N. Spaltenstein: The fixed point set of a unipotent transformation on the flag manifold, Nederl. Akad. Wetensch. Proc. Ser. A 79 (1976), [5] T.A. Springer: Trigonometric sums, Green functions of finite groups and representations of Weyl groups, Invent. Math. 36 (1976), [6] T.A. Springer: A construction of representations of Weyl groups, Invent. Math. 44 (1978), [7] T. Tanisaki: Defining ideals of the closures of the conjugacy classes and representations of the Weyl groups, Tôhoku Math. J. (2) 34 (1982), Department of Mathematics Osaka City University Sumiyoshi-ku, Osaka Japan d13sar0z06@ex.media.osaka-cu.ac.jp