Suggestions to Study Affine and GIT Quotients of the Extended

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The construction of Nakajima quiver varieties involves an action by reductive groups, but in our generalized setting, we shift our focus to parabolic group actions. Studying parabolic group actions on vector spaces with a fixed filtration and connecting affine and GIT constructions to well-known varieties, revealing different ways to view the same geometric object and often providing a deeper insight to known varieties, are important in mathematics. First, consider a classical example: let G = GLn(C) and g = gln = Lie(G). In the case of G-action on its Lie algebra g by conjugation, the Jordan quiver in §2.1.1 is the natural quiver associated to this G-equivariant geometry. The orbit space g/G is not a variety, while the affine quotient g//G consists of equivalence classes of semisimple, i.e., diagonalizable, matrices, so it is isomorphic to Cn. Algebrically, g//G := Spec(C[g]G), with the generators being coefficients of the characteristic polynomial of n×n matrices. Now, let B ≤ G be the standard Borel subgroup, i.e., the set of invertible upper triangular matrices, and let b = Lie(B). The B-action on b is naturally a quiver representation but with a notion of filtration modifying the representation space of the Jordan quiver, which one may think • n of F Rep(Q,n) to be the set of all those linear maps Wr ∈ End(C ) fixing the complete standard flag F• in Cn. The subspace F• Rep(Q,n) is acted upon by those matrices in G which preserve this flag, i.e, this is precisely B (cf. §2.1.2). Throughout this paper, we will work over C, and we will assume that the set N of natural numbers includes 0.

Acknowledgment. The author would like to thank Thomas Nevins for invaluable discus- sions in representation theory during the initial stages of this paper. The author would also like to thank Darlayne Addabbo and Bolor Turmunkh for productive research meetings to discuss topics in geometric representation theory, and, in particular, geometric constructions of numerous important algebras and on Nakajima’s work, in summer of 2014. M.S.I. was partially supported by NSA grant H98230-12-1-0216, by Campus Research Board at the University of Illinois at Urbana-Champaign, and by NSF grant DMS 08-38434.

2. Representation theory

2.1. Parabolic equivariant geometry. Studying GLn(C)-orbits on the set gln of n × n complex matrices by conjugation amounts to putting the matrices into Jordan canonical form, up to a permutation of its elementary blocks. In such area of mathematics, one hopes to construct interesting moduli spaces associated to a pair (X, G), where X is a space and G is a group. Let B be invertible upper triangular matrices in G := GLn(C) and let b be the set of upper triangular matrices in g = gln. Letting B act on b by adjoint action, we can ask what are the B-orbits on b? More generally, how does one manage a general parabolic group action on a variety that has a certain notion of filtrations associated to it? Returning to the example of the B-action on b, such pair (b,B) can be thought of as a a variety with a natural notion of filtration. The set Cn → Cn of all linear homomorphisms can be identified with g, and changing domain and codomain basis vectors amounts to a certain action by GLn(C). Now suppose we impose a complete filtration on both the domain and the codomain with respect to the standard basis vectors. That is, letting F• : 0 ⊆ C1 ⊆ C2 ⊆ . . . ⊆ Cn be the complete standard filtration of vectors, where Ck is r the subspace of Cn spanned by the first k standard basis vectors, consider the set F• → F• k k of all linear maps preserving the filtration; such maps consist of r such that r|Ck : C → C is linear for each k. Since F• is a filtration with respect to standard basis vectors, r has the form of an upper triangular matrix, and changing domain and codomain basis vectors (which preserves the filtration) amounts to an action by B. AFFINE AND GIT QUOTIENTS OF THE EXTENDED GROTHENDIECK–SPRINGER RESOLUTION 3

2.1.1. Quiver representations. We refer to [4, 12, 16] for an extensive background on quivers and their representations. Quivers Q are a directed graph with ﬁnite number of vertices and arrows. We will denote the set of vertices as Q0 and the set of arrows as Q1. An example is the Jordan quiver: 1 y • r with one vertex 1 and one arrow r; in this example, the head hr of r and the tail tr of r end and begin at the same vertex. We call the Jordan quiver cyclic since it has an oriented cycle, i.e., r is called a cycle since the tail of r equals the head of r. Another example is the A2-quiver: 1 a 2 • / • with two vertices 1 and 2 and one arrow a whose tail of a equals 1 and the head of a equals 2. We call this quiver the A2-quiver since the underlying graph has the structure of an A2-Dynkin diagram. Furthermore, we call the A2-quiver acyclic since it does not have any oriented cycles. Q0 Let v = (v1, . . . , vQ0 ) ∈ N , a dimension vector. A representation V of a quiver with dimension vector v assigns a ﬁnite dimensional vector space Vi of dimension vi to each vertex and a linear map Va to each arrow a ∈ Q1. The representation space Rep(Q, v) consists of all representations of the given quiver Q with dimension vector v. Consider the n2 Jordan quiver, and let v = n. Then Rep(Q,n) = Spec(C[Vr]) =∼ C . If v = (n,m) for the mn A2-quiver, then Rep(Q, (n,m)) = Spec(C[Va]) =∼ C . Studying isomorphism classes of representations of a quiver with a prescribed dimension vector amounts to a change-of-basis of the vector space at each vertex, i.e., it amounts to a certain natural quotient of the representation space by a group, and the study of the orbit structure is equivalent to equivariant geometry in geometric invariant theory (GIT; see §2.4).

2.1.2. Filtered ADHM data. In this section, we generalize the constructions in [33], giving a refined analogue of quiver representations called filtered quiver representations. A filtered quiver representation is a finite quiver with a fixed filtration of vector spaces at each vertex whose linear map associated to each arrow (of the quiver) preserves the filtration. Furthermore, there is a natural group action by the parabolic group Pv of Gv = ι GLvι acting on and preserving filtered vector spaces. op Q Let Q = (Q0,Q1), where cycles are allowed, and let Q1 be the same set of arrows op as Q1 but in opposite orientation. Let Q = (Q0,Q1 ⊔ Q1 ), a double quiver. Write op Q0 Q1 := Q1 ⊔ Q1 . Let v = (vι) ∈ N and let V = (Vι)ι∈Q0 be a collection of vector spaces 1 1 2 2 such that dim Vι = vι for each ι ∈ Q0. For V = (Vι )ι∈Q0 and V = (Vι )ι∈Q0 , define 1 2 1 2 1 2 1 2 L(V ,V ) := Hom(Vι ,Vι ) and E(V ,V ) := Hom(Vta,Vha). ι∈Q M0 aM∈Q1 1 2 2 3 Given B = (Ba) ∈ E(V ,V ) and C = (Ca) ∈ E(V ,V ), we also define a∈Q1 a∈Q1

1 3 CB := CaBaop ∈ L(V ,V ), ta=ι ! X ι∈Q0 where aop has the same endpoints as a but is in reverse orientation. Now, choose a sequence γ1, γ2, . . . , γN = v ∈ NQ0 of dimension vectors such that for all k k+1 vι k and ι ∈ Q0, γι ≤ γι . For each ι ∈ Q0, we get a filtration of C : 1 2 {0}⊆ Cγι ⊆ Cγι ⊆ . . . ⊆ Cvι , 4 MEE SEONG IM where each Cl is spanned by l basis elements of Cvι . Define F• Rep(Q, v, w) := F•E(V,V ) ⊕ L(W, V ) ⊕ L(V,W ), where • • F E(V,V ) := F Hom(Vta,Vha) aM∈Q1 • such that if Ba ∈ F Hom(Vta,Vha) where a ∈ Q1, then Ba preserves the fixed sequence of k k γta γ • vector spaces at every level, i.e., Ba(C ) ⊆ C ha for every k, and if Bc ∈ F Hom(Vtc,Vhc), op op • • where c = a ∈ Q1 , then F Hom(Vtc,Vhc) is the dual of F Hom(Vta,Vha) such that the trace map • • tr F Hom(Vta,Vha) × F Hom(Vtc,Vhc) −→ C, (Ba,Bc) 7→ tr(BaBc) is a nondegenerate pairing. We denote • • • C = (A, B) ∈ F E(V,V ), where A ∈ F Hom(Vta,Vha) and B ∈ F Hom(Vtc,Vhc), a∈Q op M1 c∈MQ1 i ∈ L(W, V ), and j ∈ L(V,W ). We call an element of F• Rep(Q, v, w) a filtered ADHM datum, while the filtered representation space F• Rep(Q, v, w) is called filtered ADHM data. 2.2. Invariant and semi-invariant polynomials. Invariant and semi-invariant polynomials play a fundamental role in classical and geometric invariant theory (§2.4). In fact, studying orbit spaces precisely amounts to describing invariant and semi-invariant polynomials. 2.3. Moment maps and complete intersection. Moment maps arise in symplectic geometry as a tool to construct conserved quantities. The action of a Lie group G on a vector space X is induced to the cotangent bundle T ∗X of X. Taking the derivative of the group a action induces an infinitesimal action g → T X on X, given by tangent vectors. By dualizing µ this action, one obtains the moment map T ∗X → g∗, where µ = a∗. If the zero fiber of µ is a complete intersection, then this means µ−1(0) has the expected number of irreducible components, with µ−1(0) having dimension 2 dim X − dim g. 2.3.1. Moment maps for filtered ADHM data. Recall filtered ADHM data from §2.1.2. Let ǫ : Q1 → {±1}, where

1 if a ∈ Q1, ǫ(a)= op (−1 if a ∈ Q1 . • 1 2 Then ǫC ∈ F E(V ,V ) is defined as (ǫC)a = ǫ(a)Ca for a ∈ Q1. We define a symplectic form ω on F• Rep(Q, v, w) by ′ ′ ′ ′ ′ ′ ω((C, i, j), (C , i , j )) = tr(ǫCC ) + tr(ij − i j), where tr(A)= tr(Ak). k X • Pv The product := ι∈Q0 Pvι of parabolic groups acts on F Rep(Q, v, w) via Q p ◦ (C, i, j) = (pCp−1,pi,jp−1), which preserves the symplectic form ω on the filtered ADHM data. The Pv-action induces the moment map ∗ • ∗ ∗ µPv : T F Rep(Q, v, w) → Lie(Pv) = pvι , where (C, i, j) 7→ ǫCC + ij (1) ι∈Q M0 AFFINE AND GIT QUOTIENTS OF THE EXTENDED GROTHENDIECK–SPRINGER RESOLUTION 5 and pvι = Lie(Pvι ). 2.4. Geometric invariant theory. The orbit space for a G-action on a variety X may not exist since X/G may not necessarily be an algebraic variety. In order to remedy this, invariant polynomials are used to construct new and interesting varieties called affine quotients; one may think of the affine quotient as being generated by orbit closures (or alternatively, closed orbits). Procesi in [37] proved that invariant polynomials of quiver varieties arise as traces of oriented paths in characteristic zero, and Donkin in [8] and [9] later proved an analogous result in characteristic p. To produce other interesting (and sometimes projective) varieties, one uses GIT techniques (cf. [31, 36]), where one aims to produce all semi-invariant polynomials for various character of the group. Derksen–Weyman in [6], Schofield–van den Bergh in [39], and Domokos–Zubkov in [7] used long exact sequences of the Ringel resolution, representation- theoretic techniques, and combinatorial techniques, respectively, to give a strategy to produce semi-invariant polynomials for quiver representations. Recall that given a G-action on X, affine and GIT quotients are ∞ G G,χi X//G := Spec(C[X] ) and X//χG := Proj C[X] , i ! M=0 respectively, where χ : G → C∗ is a character of G. For more detail, see [31, 36].

2.4.1. Hamiltonian reduction of filtered ADHM data. Recall µPv defined in (1). The locus µPv = 0 is called filtered ADHM equation. We say • • F F −1 −1 Pv v M0 = M0 (v, w) := µPv (0)//P = Spec C[µPv (0)] is the filtered affine quotient, and

• • i F F −1 s −1 Pv ,χ M = M (v, w) := µ (0) /Pv = Proj C[µ (0)] χ χ Pv  Pv  i≥ M0 is called a ﬁltered quiver variety.   In the case when Q is the Jordan quiver 1 y • r (2) then the double quiver Q is 1. y op r 9 • r (3) Let v = n and w = 1. We impose the complete standard ﬁltration of vector spaces on the n −1 ∼ ∗ n representation V1 = C at vertex 1 to obtain that µB (0)/B = T (g × C / GLn(C)) (cf. [35, Prop. 3.2], [17, Prop. 1.1]). e 3. Invariants and semi-invariants of filtered quiver representations We obtain the following in [16, Thm. 5.1.2]:

Theorem 3.1 (Im). Consider a quiver Q = (Q0,Q1) of finite Dynkin type with dimension vector v = (n,...,n). Impose the complete standard filtration of vector spaces on the representation at each vertex of the quiver. Let U be the product of maximal unipotent subgroups of the product B = BQ0 of standard Borels. Then C[bQ1 ]U =∼ C[tQ1 ]. Theorem 3.1 says that invariant polynomials cannot arise from off-diagonal coordinates of b. Instead, only the eigenvalues of bQ1 remain invariant under the U-action. Next, we state [16, Thm. 5.2.12]. 6 MEE SEONG IM

Theorem 3.2 (Im). Consider an affine quiver Ar with a framing, and let v = (n,...,n,m), the dimension vector. Assume the complete standard filtration of vector spaces on the representation at each vertex of Ar (except at the framede vertex). Let U be the product of maximal unipotent subgroups of the product B = Br of standard Borels. Then the invariant subring ⊕r+1 U C[b ⊕ Mn×m] has finitely-manye generators. We give an explicit description of the subalgebra in Theorem 3.2, thus generalizing [13, Thm. 13.3], which states that for the maximal unipotent group U of the standard Borel U B ⊆ GLn(C), the algebra C[Mn×m] is generated by bideterminants of standard Young bitableaux (D|E), where each row of D has the form p,p+1,...,n for a suitable p, 1 ≤ p ≤ n. Note that one may associate a bideterminant to a Young bitableau in a natural way, i.e., it is a product of minors, each of which is determined by i-th row of D and i-th row of E. Also see [15, Thm. 1.1, Thm. 1.3] as further extensions of Theorem 3.1 and Theorem 3.2.

4. Hamiltonian reduction of the Borel moment map Throughout this section, we will restrict to the Jordan quiver. Let v = n and w = 1. Let F• be the complete standard ﬁltration of vector spaces. Then F• Rep(Q,n, 1) = T ∗(b × Cn), which is associated to the moment map

∗ n ∗ µB : T (b × C ) → b , where (r, s, i, j) 7→ [r, s]+ ij.

−1 rss Let µB (0) be the locus of points whose eigenvalues of r are pairwise distinct. Let ∆n = {(r1, . . . , rn, 0,..., 0) : rι = rγ for some ι 6= γ}. Then [17, Thm. 1.5] says that: Theorem 4.1 (Im). The map −1 rss ։ 2n ′ ′ P : µB (0) C \ ∆n given by (r, s, i, j) 7→ (r11, . . . , rnn,s11,...,snn) is a regular, well-deﬁned surjection separating B-orbit closures.

Furthermore, we have [17, Thm. 1.6]:

Theorem 4.2 (Im). The surjection P in Theorem 4.1 descends to an isomorphism −1 rss ∼ 2n µB (0) //B = C \ ∆n of algebraic varieties.

We prove that the components of µB form a regular sequence using a certain monomial ordering and the resulting initial ideal. In the process of proving Theorem 4.1 and Theorem 4.2, rational B-invariant polynomials appearing as traces of products of matrices were discovered (cf. [16, §6.2]): for 1 ≤ ι ≤ n and 1 ≤ γ<ν ≤ n, ι Fι(r, s, i, j) = tr (jL i) , ι Gι(r, s, i, j) = tr (L s) , ι (4) Hι(r, s, i, j) = tr(L r), ν γ −1 Kγν (r, s, i, j) = [tr((L − L )r)] .

These rational functions should play an important role in the construction of the aﬃne −1 quotient µB (0)//B (see §6.2 for more detail). AFFINE AND GIT QUOTIENTS OF THE EXTENDED GROTHENDIECK–SPRINGER RESOLUTION 7

5. Hamiltonian reduction of the parabolic moment map In this section, we continue to work with the Jordan quiver. Let v = n and w = 1. Here, rather than working with a complete ﬂag as in §4, we work with a partial standard ﬂag F• : Cα1 ⊆ Cα1+α2 ⊆ . . . ⊆ Cα1+...+αℓ , where ℓ ≤ 5. (5) Then we have [18, Thm. 1.1]:

Theorem 5.1 (Im–Scrimshaw). Let α = (α1,...,αℓ) such that ℓ ≤ 5. Let P be the parabolic subgroup of GLn(C) with block size vector α. The components of µP form a complete intersection. We also explicitly describe the irreducible components in [18, Thm. 1.2].

6. Open problems We will now list some research problems related to ﬁltered ADHM data. 6.1. (Semi-)invariant polynomials for a general quiver. As a result of the B-invariant functions in (4), let us now focus on the n rational functions Lι enumerated by ι = 1,...,n. For lk(r)= r − rkk In,

M ι := tr l (r) Lι = l (r),   k  k ≤k≤n,k6 ι ≤k≤n,k6 ι 1 Y = 1 Y = which is an operator whose matrix entries are zero except in coordinates (µ,ν) for µ ≤ ι and ν ≥ ι. One should think of these operators acting on elements in b as killing off columns ∗ + of a matrix, or as creating new coordinate entries from matrices in b = gln/u such that powers of the trace of the product of these matrices give the desired invariant polynomials for the filtered affine quotient setting. Problem 6.1. Describe C[b⊕k]Un×Un for k ≥ 3, and C[b⊕ℓ]Un for ℓ ≥ 2. By making progress on Problem 6.1, together with Theorem 3.2, one can then describe the unipotent invariant subring for a filtered ADHM data for any quiver Q. 6.2. Affine and GIT quotients. Let Q be the Jordan quiver. By clearing the denomina- −1 tors of the rational functions in (4), they are some of the generators of C[µB (0)//B]. More generally, we have: −1 −1 v v Problem 6.2. Find generators and relations for µPv (0)//P and µPv (0)//χP to describe interesting quotients. Problem 6.2 is of interest for constructing new moduli spaces, and it remains an open problem to study the algebro-geometric structure of these quotients for various quivers, filtrations, and stability conditions, i.e., for different characters. 6.3. Birational morphisms. −1 −1 v ′ v Problem 6.3. Construct birational morphisms between µPv (0)//χP and µPv (0)//χ P for two characters χ and χ′ of Pv. Problem 6.3 is known as variational GIT. Fixing a character of the group and constructing GIT quotients is known as a polarization on the variety, and changing one polarization to another is known as wall-crossing. Since a parabolic group has many more characters than GLn(C), this problem now becomes even more interesting than a reductive group setting. 8 MEE SEONG IM

Thus what happens when we fix a character of the parabolic group and the phenomenon that arises when we cross a wall? Are the resulting GIT quotients birational or isomorphic? Even when we restrict the quiver to the Jordan quiver, studying variational GIT and constructing Fourier automorphism, thus giving an isomorphism between certain open loci of GIT quotients, are doable yet a difficult task (one needs to choose various 1-parameter subgroups and show that certain points are unstable with respect to a fixed character).

−1 6.4. Complete intersection for µP (0). It remains to prove that when ℓ > 5 for ℓ in −1 (5), the associated parabolic moment map is flat, i.e., µP (0) is a complete intersection or equivalently, the components of µP form a regular sequence. Furthermore, one should use −1 n the well-known fact that µP is flat if and only if dim µP (0)//P = 2(dim(p × C ) − dim p) if and only if n codim{y ∈ p × C : dim Gy = k}≥ k for all k ≥ 1. Results in [17] coincide with the classical notion that the trace of an oriented cycle of a quiver, as well as the trace of a path that begin and end at a framed vertex, is an invariant function (cf. [5, 22, 27]). These strategies are appliable to Nakajima’s affine and quiver −1 P varieties. This leads us to believe, together with [10, Proof of Prop. 8.2.1], that C[µP (0)] is generated by traces of products of matrices.

Problem 6.4. The filtered ADHM equation µP is a complete intersection for ℓ> 5. 6.5. Complete intersection for the general quiver. Regular sequences for the filtered ADHM equation play an important role in affine and GIT quotients. A set of polynomials f1,...,fN is C[x1,...,xm]-regular if the scheme defined by the vanishing locus of f1,...,fN form a complete intersection. That is, the variety has the expected dimension m − N, and in such a case, we say that the components form a regular sequence. The affine quotient of a variety is generated by invariant polynomials, and GIT quotients are generated by semi- invariant polynomials. In the case that the components of a moment map form a regular sequence, the affine and the GIT quotients are finitely generated; this means we only need to produce finitely-many invariant and semi-invariant polynomials since the quotients will have the expected dimension.

Problem 6.5. Describe when components of µPv form a regular sequence. 6.6. Comparing ﬁltered ADHM data and quivers with relations. There is a close relation to quivers with relations but examples show that they are not equivalent. It would be interesting to check under what conditions are irreducible components of quivers with relations normal. In the case that components of quivers with relations are normal, the function theory for quivers with relations can be extended to all of the component, and, thus, is comparable to ﬁltered ADHM data.

6.7. Hilbert schemes. Writing T ∗(b × Cn) as b × b∗ × Cn × (Cn)∗, it contains the set of all quadruples of the form (r, s, i, j), where s takes the form of lower triangular matrices ∗ (technically, b = gln/u where u is the set of nilpotent matrices in b), i is a vector, and j is a covector. Framing means that there is no group action on the vector space at that vertex. Restricting to those points satisfying the relation [r, s]+ ij = 0 should remind the experts of the Hilbert scheme (C2)[n] of n points on the complex plane. −1 ։ −1 Problem 6.6. Construct µP (0)//χP µP (0)//P and relate it to the Hilbert–Chow mor- 2 [n] 2n phism (C ) ։ C /Sn, where Sn is a permutation group of n letters. AFFINE AND GIT QUOTIENTS OF THE EXTENDED GROTHENDIECK–SPRINGER RESOLUTION 9

6.8. Idempotents in ﬁltered quiver representations and quivers with relations. The Lι’s in [17] (also see [16]) are n rational idempotents. They are pairwise orthonormal and their sum equals the identity matrix. These idempotents are all upper triangular matrices whose coordinate are rational functions. In §6.6, we compare ﬁltered ADHM data with quivers with relations. In view of the Borel subalgebra corresponding to the Jordan quiver embedded in a quiver with relations, whose quiver has n vertices and each vertex has a trivial path (also known as idempotents in its path algebra), it is highly likely that these idempotent matrices are directly related to the n idempotents appearing in quivers with relations.

6.9. Modiﬁed rational Cherednik algebras. In [10], the authors consider the spherical subalgebra of gln-type and realize it as a quantum Hamiltonian reduction of the algebra D(g) of polynomial diﬀerential operators on g. Since studying the Hamiltonian reduction of D(g) with respect to P is the same as performing the Hamiltonian reduction of D(g × Pn) with respect to G (acting diagonally on g × Pn), this leads us to an analogous investigation for parabolic groups:

Problem 6.7. Construct a parabolic analog of the Cherednik algebra and realize it as a quantization of D(p).

6.10. Parabolic D-modules. In a similar spirit as in [19], we believe that the quantization −1 of the Hilbert scheme associated to µP (0)//χP may be realized as a microlocalization of a modiﬁed rational Cherednik algebra. Furthermore, Gan–Ginzburg construct a Lagrangian subvariety Mnil in M in the classical case of GLn(C), and then studies a category of holo- nomic D-modules whose characteristic variety is contained in Mnil. Problem 6.8. Construct a category of D-modules whose characteristic variety is contained P in the parabolic analogue Mnil and compare its simple objects to Lusztig’s (parabolic) character sheaves (cf. [30, 23]).

−1 6.11. Derived category of coherent sheaves on µP (0)//χP . There is an equivalence between derived categories of coherent sheaves on (C2)[n] and finitely-generated modules 2n over a noncommutative crepant resolution of C /Sn (cf. [14]). Problem 6.9. Construct an equivalence between derived categories of coherent sheaves on −1 µP (0)//χP and the category of finitely-generated modules over a modified rational Cherednik algebra.

6.12. Generalized Grothendieck–Springer resolution. It is a classical result in representation theory that the Grothendieck–Springer resolution g ։ g, where (x, b) 7→ x, is generically ﬁnite and dominant of degree |W |, where W is the Weyl group. e Problem 6.10. Extend µPv from the Jordan quiver to a general quiver as it is interesting to study the resolution from a ﬁltered quiver variety point of view.

Directly studying the parabolic orbits on the Grothendieck–Springer resolution is very difficult, and surprisingly, no one seems to have studied such geometric objects from the quiver variety point of view. Filtered ADHM data represent a new technique for studying objects in geometric representation theory and algebraic geometry. They are nice because of their concrete nature and because of many applications in mathematics. Using filtered quiver variety techniques, we hope that new and different insights to important varieties will be identified and discovered. 10 MEE SEONG IM

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Department of Mathematical Sciences, United States Military Academy, West Point, NY 10996 USA E-mail address: [email protected]