Computational methods in the study of symplectic quotients

Transcription

1 Computational methods in the study of symplectic quotients Hans-Christian Herbig, UFRJ and Christopher Seaton, Rhodes College Instituto de Matemática Aplicado, Universidade Federal do Rio de Janeiro January th, 2016 UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

2 Minicourse Abstract Let G be a compact Lie group and let V be a unitary G-representation. Then there is a quadratic moment map J : V g with respect to which V is a Hamiltonian manifold. Letting Z denote the zero fiber J 1 (0) of the moment map, the corresponding symplectic quotient is given by M 0 = Z/G. It has the structure of a symplectic stratified space as well as a semialgebraic set, and it is equipped with an algebra of regular functions R[M 0 ], a Poisson subalgebra of its algebra of smooth functions. In these lectures, we will introduce methods of computing the algebra R[M 0 ] of regular functions on such a symplectic quotient using methods from invariant theory and computational algebraic geometry. In addition, we will explain how these computations can be used to observe and verify properties of the symplectic quotient. Topics will include using Groebner bases to compute invariant polynomials, elimination theory, and methods of computing Hilbert series of Cohen-Macaulay algebras. In addition, we will introduce the software packages Mathematica and Macaulay2 for these kinds of computations. UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

3 Minicourse Lectures 1 Invariant theory and Gröbner bases 2 Singular symplectic reduction and regular functions on symplectic quotients 3 The Hilbert series of the regular functions on a symplectic quotient 4 Elimination theory and the nonabelian case UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

4 Lecture 1: Invariant Theory and Groebner Bases 1 Affine varieties and ideals of polynomials 2 Invariant Polynomials 3 Gröbner Bases 4 Representations of Tori 5 Finding invariants using Gröbner bases 6 Finding Relations Using Gröbner Bases UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

5 Affine varieties and ideals of polynomials Affine varieties and ideals of polynomials UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

6 Affine varieties and ideals of polynomials Affine Varieties Let K denote either R or C. To fix notation: K n is affine space, the vector space {(a 1,..., a n ) : a i K} under component-wise addition and K-multiplication. A monomial in the variables x 1,..., x n is an expression of the form x p 1 1 x p 2 2 x n pn where each p i is a nonnegative integer. The degree of x p 1 1 x p 2 2 x n pn is p p n. K[x 1,..., x n ] is the set of polynomials in x 1,..., x n with coefficients in K, i.e. finite linear combinations of monomials in x 1,..., x n. We identify K[x 1,..., x n ] with a subset of the continuous functions K n K in the obvious way. A subset V of K n is an affine variety if there is a finite subset F K[x 1,..., x n ] such that V can be described as V = V(F) := {(a 1,..., a n ) K n : f (a 1,..., a n ) = 0 f F}. UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

7 Affine varieties and ideals of polynomials Affine Varieties: Examples Example In R 2, the unit circle is the vanishing set of F = {x 2 + y 2 1}, hence an affine variety UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

8 Affine varieties and ideals of polynomials Affine Varieties: Examples Example In R 2, the variety corresponding to F = {x 2 y 2 } consists of the lines y = x and y = x UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

9 Affine varieties and ideals of polynomials Affine Varieties: Examples Example In R 3, the variety corresponding to F = {x 2 + y 2 z} is a paraboloid: UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

10 Affine varieties and ideals of polynomials Affine Varieties: Examples Example In R 3, the variety corresponding to F = {x 2 + y 2 z 2 } is a cone: UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

11 Affine varieties and ideals of polynomials Affine Varieties: Examples Example In R 3, the variety corresponding to F = {x 2 + y 2 z 2, x 2 + y 2 z} the intersection of the cone and paraboloid: UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

12 Affine varieties and ideals of polynomials Affine Varieties: Examples Example In R 3, the Whitney umbrella is the variety of F = {x 2 y 2 z}: UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

13 Affine varieties and ideals of polynomials Ideals: Motivation Let V = V(F) be the variety of K n described by the subset F K[x 1,..., x n ]. If f F and g K[x 1,..., x n ], then (fg)(a 1,..., a n ) = 0 (a 1,..., a n ) V. For instance, in R[x, y], any polynomial of the form (x 2 + y 2 1)g(x, y) must vanish on the unit circle. Hence, the subset of K[x 1,..., x n ] that vanishes on V is much larger than F. UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

14 Affine varieties and ideals of polynomials Ideals Definition Let I be a subset of K[x 1,..., x n ]. Then I is an ideal if 0 I, f 1, f 2 I, f 1 + f 2 I, and f I, g K[x 1,..., x n ], fg I. If f 1,..., f r K[x 1,..., x n ], the ideal generated by f 1,..., f r is { r } f 1,..., f r = h i f i : h i K[x 1,..., x n ]. i=1 It is the smallest ideal containing f 1,..., f r. Exercise: Show that f 1,..., f r is in fact an ideal. UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

15 Affine varieties and ideals of polynomials Ideals Definition Let S be any subset of K n. The ideal of S is I(S) = {f K[x 1,..., x n ] : f (c 1,..., c n ) = 0 (c 1,..., c n ) S}. i.e. the set of polynomials that vanish on S. To see that I(S) is in fact an ideal, let S be an arbitrary subset of K n. 0 vanishes on all of K n so 0 I(S) is obvious. If f 1, f 2 I(S), then for each (c 1,..., c n ) S, f 1 (c 1,..., c n ) = f 2 (c 1,..., c n ) = 0.Therefore (f 1 + f 2 )(c 1,..., c n ) = = 0 and f 1 + f 2 I(S). If f I(S) and g K[x 1,..., x n ], then for each (c 1,..., c n ) S, f (c 1,..., c n ) = 0.Therefore (fg)(c 1,..., c n ) = f (c 1,..., c n )g(c 1,..., c n ) = 0 g(c 1,..., c n ) = 0 and fg I(S). UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

16 Affine varieties and ideals of polynomials Ideals and Varieties So we have: subsets of K n I ideals of K[x 1,..., x n ] and subsets of K[x 1,..., x n ] V varieties in K n. Lemma If f 1,..., f r K[x 1,..., x n ], then f 1,..., f r I(V(f 1,..., f r )). UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

17 Affine varieties and ideals of polynomials Ideals and Varieties Proof. Given r i=1 h i f i f 1,..., f r, pick (c 1,..., c n ) V(f 1,..., f r ). Then for each i, f i (c 1,..., c n ) = 0 by definition. Hence ( r ) r h i f i (c 1,..., c n ) = h i (c 1,..., c n )f i (c 1,..., c n ) So r i=1 i=1 h i f i I(V(f 1,..., f r )). = i=1 r h i (c 1,..., c n ) 0 = 0. i=1 UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

18 Affine varieties and ideals of polynomials Ideals and Varieties However, I(V(f 1,..., f r )) is often larger than f 1,..., f r. Example In R[x], the ideal I = x 2 contains all polynomials with no constant or linear terms. Then V(I ) = {0}. However, I(V(I )) contains x. UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

19 Affine varieties and ideals of polynomials Ideals and Varieties Lemma If S K n, then S V(I(S)). Exercise: Prove this lemma. Again, equality need not hold. Example If S = Q R, any function that vanishes on Q must vanish on R by continuity, so I(S) = {0} and V(I(S)) = R. UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

20 Affine varieties and ideals of polynomials Radical Ideals Definition (K = C) The radical of an ideal I of C[x 1,..., x n ] is I := {f C[x1,..., x n ] : f m I for some m > 0}. An ideal I of C[x 1,..., x n ] is radical if I = I, i.e. for any f C[x 1,..., x n ], if f m I for a positive integer m, then f I. (K = R) The real radical of an ideal I of R[x 1,..., x n ] is R I := {f R[x1,..., x n ] : g 1,..., g r R[x 1,..., x n ], f 2m + g g 2 r I for some m > 0} An ideal I of R[x 1,..., x n ] is real if I = R I, i.e. for any f 1,..., f r R[x 1,..., x n ], f f r 2 I for a positive integer m implies f 1,..., f r I. UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

21 Affine varieties and ideals of polynomials Correspondence between Ideals and Varieties Theorem (Hilbert s Nullstellensatz) If I is an ideal of C[x 1,..., x n ], then I(V(I )) = I. Hence, there is a bijection between affine varieties in C n and radical ideals in C[x 1,..., x n ]. If I is an ideal of R[x 1,..., x n ], then I(V(I )) = R I. Hence, there is a bijection between affine varieties in R n and real ideals in C[x 1,..., x n ]. Proofs can be found in Cox Little O Shea [2] (over C) and Bochnak Coste Roy [1] (over R). In the correspondence, the ideals of K[x 1,..., x n ] that are maximal (contained in no larger ideal except K[x 1,..., x n ] itself) correspond to points in the variety. UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

22 Affine varieties and ideals of polynomials The Polynomial Functions on a Variety If I is an ideal of K[x 1,..., x n ], define the equivalence class K[x 1,..., x n ] by mod I on g 1 g 2 mod I iff g 1 g 2 f. The equivalence class of g is denotes g + I. The quotient algebra K[x 1,..., x n ]/I is defined to be the set of equivalence classes in K[x 1,..., x n ]. It can be shown that the operations (g 1 + I ) + (g 2 + I ) := (g 1 + g 2 ) + I and (g 1 + I )(g 2 + I ) := (g 1 g 2 ) + I are well defined on K[x 1,..., x n ]/I. If I = I(V ) is the ideal of a variety V, then K[x 1,..., x n ]/I is thought of as the polynomial functions on V. Two elements of K[x 1,..., x n ] represent the same element of K[x 1,..., x n ]/I if and only if they have the same value at every point in V. UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

23 Invariant Polynomials Invariant polynomials UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

24 Invariant Polynomials Invariant Polynomials Let GL n (K) denote the group of invertible n n matrices with entries in K. Definition A subset G GL n (K) is a subgroup, written G GL n (K), if The identity Id G, If A, B G, then the matrix product AB G, and If A G, then A 1 G. If, G GL n (K) and f K[x 1,..., x n ], then f is G-invariant if f A = f for each A G. The collection of all G-invariant polynomials is denoted K[x 1,..., x n ] G. UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

25 Invariant Polynomials Invariant Polynomials: Basic Examples Let G = {1, 1} GL 1 (R). For f R[x], it is easy to see that f (x) = f ( x) if and only if f (x) is even, i.e. f (x) = g(x 2 ) for some g R[w]. Hence, R[x] G = {g(x 2 ) : g R[w]}. UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

26 Invariant Polynomials Invariant Polynomials: Basic Examples Let A = ( ) 0 1, and let G = {Id, A} GL (C). For f C[x, y], we have (f A)(x, y) = f (y, x). Exercise: For f C[x, y], f A = f if and only if f (x, y) = g(x + y, xy) for some g C[w 1, w 2 ]. Hint: If h is a monomial of degree d, then g A is a monomial of degree d. So f is G-invariant if and only if it is the sum of homogeneous G-invariant polynomials (i.e. all terms have the same degree). Hence, C[x, y] G = {g(x + y, xy) : g C[w 1, w 2 ]}. Elements of C[x, y] G are called symmetric polynomials in two variables. UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

27 Invariant Polynomials Hilbert Bases The set K[x 1,..., x n ] G is closed under addition, multiplication, and scalar multiplication of polynomials, and hence is a subalgebra or subring of K[x 1,..., x n ]. This is easy to see, e.g. (f + g) A = (f A) + (g A) so that (f A) = f and (g A) = g implies (f + g) A = f + g. Note that K[x 1,..., x n ] G is not an ideal of K[x 1,..., x n ]. We refer to {g(x 2 ) : g C[w]} as the subalgebra generated by x 2, written C[x 2 ]. Similarly, {g(x + y, xy) : g C[w 1, w 2 ]} = C[x + y, xy] is the subalgebra generated by {x + y, xy}. In general, the subalgebra generated by f 1,..., f r K[x 1,..., x n ] is {g(f 1,..., f r ) : g K[w 1,..., w r ]}. We refer to {f 1,..., f r } as a Hilbert basis for the subalgebra. Hilbert basis (even minimal Hilbert bases) are often not unique. UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

28 Invariant Polynomials Invariant Polynomials: Another Example ( ) 1 0 Let B =, and let G = {Id, B} GL (R). For f R[x, y], we have (f B)(x, y) = f ( x, y). Hence, each monomial of degree d is multiplied by ( 1) d. For f R[x, y], f B = f if and only if f (x, y) = g(x 2, y 2, xy) for some g R[w 1, w 2, w 3 ]. Hence, {x 2, y 2, xy} is a Hilbert basis for R[x, y] G, i.e. R[x, y] G = R[x 2, y 2, xy]. In this example, however, the elements of the Hilbert basis satisfy a relation: (x 2 )(y 2 ) = (xy) 2. UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

29 Invariant Polynomials Algebraic Dependence Definition We say that a finite set {f 1,..., f r } K[x 1,..., x n ] is algebraically independent if g(f 1,..., f r ) = 0 implies g = 0. If there is a nonzero g K[w 1,..., w r ] such that g(f 1,..., f r ) = 0, then {f 1,..., f r } is algebraically dependent. If a subalgebra has an algebraically independent Hilbert basis {f 1,..., f r }, then the subalgebra has the same properties as K[w 1,..., w r ]. We can think of it as the same as polynomial functions on K r. UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

30 Invariant Polynomials Algebraic Dependence If {f 1,..., f r } is algebraically dependent, define R(f 1,..., f r ) = {g K[w 1,..., w r ] : g(f 1,..., f r ) = 0}. Then R(f 1,..., f r ) is an ideal, the ideal of relations of {f 1,..., f r }. It is easy to see that R(f 1,..., f r ) is an ideal: Obviously, 0 R(f 1,..., f r ). If g 1, g 2 R(f 1,..., f r ), then g 1 (f 1,..., f r ) = g 2 (f 1,..., f r ) = 0, so (g 1 + g 2 )(f 1,..., f r ) = = 0 and (g 1 + g 2 ) R(f 1,..., f r ). If g R(f 1,..., f r ) and h K[w 1,..., w r ], then (gh)(f 1,..., f r ) = g(f 1,..., f r )h(f 1,..., f r ) = 0 h(f 1,..., f r ) = 0, and gh R(f 1,..., f r ). If a subalgebra has an algebraically dependent Hilbert basis {f 1,..., f r }, then the subalgebra is the same as the polynomial functions on the variety V(R(f 1,..., f r )). UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

31 Invariant Polynomials Invariant Polynomials: Example Revisited For B = ( ) 1 0 and G = {Id, B} GL (R), R[x, y] G = R[x 2, y 2, xy]. The Hilbert basis {x 2, y 2, xy} is algebraically dependent, with relation (x 2 )(y 2 ) = (xy) 2, i.e. w 1 w 2 w 2 3 = 0. The ideal of relations is R(x 2, y 2, xy) = w 1 w 2 w 2 3. Hence, R[x, y] G is the same as the polynomial functions on the affine variety in R 3 defined by w 1 w 2 w 2 3. UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

32 Invariant Polynomials Invariant Polynomials: Example Revisited The affine variety in R 3 defined by w 1 w 2 w 2 3 : UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

33 Invariant Polynomials Invariant Polynomials: Some Facts For an arbitrary subgroup G GL n (K), K[x 1,..., x n ] G might not have a Hilbert basis. For many subgroups (reductive subgroups) it does. If G is a closed subgroup of GL n (K) then it is an example of a Lie group. Compact (bounded) Lie groups are always reductive by a theorem of Hilbert and Weyl. If {f 1,..., f r } is a Hilbert basis for K[x 1,..., x n ] G, the variety V(R(f 1,..., f r )) plays the role of the quotient of K n by G, and is written K n /G. It is called the affine GIT quotient. When K = C, there is a bijection between the closed G-orbits in C n and C n /G. When K = R, there is a bijection between the closed G-orbits in K n and a subset of R n /G. When G is a compact Lie group, all orbits are closed. UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

34 Invariant Polynomials An Example with Non-Closed Orbits Let G = GL 1 (C). This is the group of nonzero complex numbers, denoted C. For f C[x], it is easy to see that f (x) = f (zx) z C if and only if f (x) = c for some c C. That is, C[x] C = C. Hence, C[x] C is the polynomial functions on a point (no variables). In terms of this action, C has two orbits: {0} and all other points. The variety of C[x] C, a single point, corresponds to the closed orbit {0}. UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

35 Gröbner Bases Gröbner Bases UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

36 Gröbner Bases Ideal Membership Ideal Membership Problem: Given an ideal I = f 1,..., f r K[x 1,..., x n ] and a polynomial g K[x 1,..., x n ], decide whether g f 1..., f r. If n = r = 1, then we can use polynomial division to find the answer. Example In R[x], the polynomial x 5 3x is not an element of x This can be seen by dividing x 5 3x by x and seeing that the remainder is x + 5. UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

37 Gröbner Bases Ideal Membership Example In R[x], the polynomial x 5 3x 2 x 3 is an element of x Dividing x 5 3x 2 x 3 by x 2 + 1, we see that x 5 3x 2 x 3 = (x 3 x 3)(x 2 + 1) x In fact, it can be shown that when n = 1 (one variable), every ideal is of the form f, so r = 1 in every case. UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

38 Gröbner Bases Ideal Membership Ideal Membership Problem: Given an ideal I = f 1,..., f r K[x 1,..., x n ] and a polynomial g K[x 1,..., x n ], decide whether g f 1..., f r. A natural idea to try is to divide g by f 1, then the remainder by f 2, then the remainder by f 3, etc. to try to express g in the form r i=1 h if i for some h i K[x 1,..., x n ]. But polynomial division depends on how you order the terms of g, and if we fix this, the answer depends on the order of f 1,..., f r, etc. UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

39 Gröbner Bases Monomial Orders Definition A monomial order on K[x 1,..., x n ] is a linear order on the set of monomials such that Example 1 m for each monomial m, and m 1 m 2 implies m 1 m 3 m 2 m 3 for monomials m 1, m 2, m 3. In K[x], the only monomial order is 1 x x 2 x 3. With more than one variable, it is typical to assume that x 1 x 2. Example Lexicographic order: m 1 = x p 1 1 x n pn m 2 = x q 1 1 x n qn nonzero entry of p 1 q 1, p 2 q 2,..., p n q n is negative. if the first UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

40 Gröbner Bases Monomial Orders Example Degree lexicographic order: m 1 = x p 1 1 x n pn m 2 = x q 1 1 x n qn if deg m 1 < deg m 2 or deg m 1 = deg m 2 and the first nonzero entry of p 1 q 1, p 2 q 2,..., p n q n is negative. Example Degree reverse lexicographic order: m 1 = x p 1 1 x n pn m 2 = x q 1 1 x n qn if deg m 1 < deg m 2 or deg m 1 = deg m 2 and the last nonzero entry of p 1 q 1, p 2 q 2,..., p n q n is positive. UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

41 Gröbner Bases Gröbner Bases Definition Given a monomial order on K[x 1,..., x n ], let f K[x 1,..., x n ], and let I be an ideal of K[x 1,..., x n ]. The initial monomial init(f ) of f is the largest monomial in f with respect to. The initial ideal of the ideal I is the ideal generated by the initial monomials of the elements of I. A finite set {g 1,..., g s } of I is a Gröbner basis for I if {init(g 1 ),..., init(g s )} generates the initial ideal of I. A Gröbner basis {g 1,..., g s } for I is reduced if, for i j, init(g i ) does not divide any monomial in g j. UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

42 Gröbner Bases Properties of Gröbner Bases Gröbner bases have properties that can be used to solve problems like the Ideal Membership Problem. There is a division algorithm that allows one to divide a polynomial g by a Gröbner basis for an ideal I, yielding a unique remainder. (The remainder is zero if and only if g I ). Gröbner bases generalize the Euclidean algorithm for polynomials to the multivariable case, and Gaussian elimination to polynomials of degree larger than 1. Buchberger s algorithm is an algorithm for computing a Gröbner basis for an ideal of K[x 1,..., x n ], and is implemented on many computer algebra systems. More information can be found in Cox Little O Shea [2]. UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

43 Gröbner Bases Computing Gröbner bases on Mathematica To compute the Gröbner basis of the ideal I = f 1,..., f r in K[x 1,..., x n ] on Mathematica, the command is GroebnerBasis[{f1,f2,...,fr}, {x1,x2,...,xr}] The monomial order is lexicographic (and based on the order in which the variables are listed). GroebnerBasis[{f1,f2,...,fr}, {x1,x2,...,xr}, MonomialOrder->DegreeReverseLexicographic] changes the monomial order. GroebnerBasis[{f1,f2,...,fr}, {x1,x2} {x1,x2,...,xr}, eliminates x 1 and x 2 in the Gröbner basis (i.e. removes any elements involving these variables). UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

44 Representations of Tori Representations of Tori UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

45 Representations of Tori Compact Tori Let S 1 = {z C : z = 1}, considered with the operation of multiplication. This is the unit circle in the complex plane, and it is closed under multiplication and inversion. We define the l-dimensional (compact) torus to be with the operation T l := (S 1 ) l (t 1,..., t l ) (t 1,..., t l ) = (t 1t 1,..., t l t l ). UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

46 Representations of Tori Compact Tori as Subgroups of GL n (C) There are many subgroups of GL n (C) that can be identified with T l. T 1 = S 1 is a subgroup of GL 1 (C) = C. We can identify T 1 with {( ) } t 0 : t T 1 GL 0 t 2 (C). We can also identify T 1 with {( ) } t t 2 : t T 1 GL 2 (C). We can identify T 2 with {( t1 0 0 t 1 1 t 2 ) } : (t 1, t 2 ) T 2 GL 2 (C). UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

47 Representations of Tori Compact Tori as Subgroups of GL n (C) A weight matrix A is an l n matrix with integer entries. It describes a specific subgroup of GL n (C) that can be identified with the torus T r where r is the rank of A (which we usually assume is l). The subgroup is given by diagonal matrices with diagonal entries (t a 1,1 1 t a 2,1 2 t a l,1 l, t a 1,2 1 t a 2,2 2 t a l,2 l,..., t a 1,n 1 t a 2,n 2 t a l,n l ) where (t 1,..., t l ) T l. Note that Gaussian elimination (over Z) and permuting columns of the weight matrix doesn t really change the subgroup, just expresses it using different coordinates. Up to equivalence, every subgroup of GL n (C) that can be identified with T l can be expressed by a weight matrix. UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

48 Representations of Tori Compact Tori as Subgroups of GL n (C), Examples The weight matrix (1) describes T 1 as it is defined in GL 1 (C), i.e. matrices (t) with t = 1. The weight matrix ( 2, 3) describes T 1 as the subgroup {( ) } t t 3 : t T 1 GL 2 (C). The weight matrix ( ) 2 3 describes T as the subgroup {( ) } t t1 3 : (t 1, t 2 ) T 2 GL 2 (C). t4 2 UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

49 Representations of Tori Invariants of Compact Tori Let A be an l n weight matrix and G the corresponding subgroup of GL n (C). A polynomial f C[x 1,..., x n ] is G-invariant if and only if each of its monomial terms is invariant. The monomial x p 1 1 x p 2 2 x n pn is invariant if and only if p 1 p 2 A. = 0. p n UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

50 Representations of Tori Invariants of Compact Tori, Examples For A = ( ), the monomial f (x 1, x 2 ) = x1 2x 2x 3 is invariant, as, for any t 1 T 1, f (t 1 1 x 1, t 1 x 2, t 1 x 3 ) = (t 1 1 x 1) 2 (t 1 x 2 )(t 1 x 3 ) = x 2 1 x 2 x 3 = f (x 1, x 2, x 3 ). The monomial f (x 1, x 2 ) = x 2 x 3 is not invariant, as unless t 1 = 1. f (t 1 1 x 1, t 1 x 2, t 1 x 3 ) = (t 1 x 2 )(t 1 x 3 ) = t 2 1x 2 x 3 f (x 1, x 2, x 3 ) UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

51 Representations of Tori Invariants of Compact Tori, Examples A = ( ) A Hilbert basis for C[x 1, x 2, x 3 ] G is given by {x 1 x 2, x 1 x 3 }, i.e. C[x 1, x 2, x 3 ] G = C[x 1 x 2, x 1 x 3 ]. UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

52 Representations of Tori Invariants of Compact Tori, Examples ( ) For A =, the monomial f (x , x 2, x 3 ) = x 1 x2 3x 3 2 as, for any (t 1, t 2, t 3 ) T 2, is invariant, f (t 2 1 t 3 2 x 1, t 2 x 2, t 1 x 3 ) = (t 2 1 t 3 2 x 1)(t 2 x 2 ) 3 (t 1 x 3 ) 2 = x 1 x 3 2 x 2 3 = f (x 1, x 2, x 3 ). A Hilbert basis for C[x 1, x 2, x 3 ] G is given by {x 1 x 3 2 x 2 3 }. UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

53 Representations of Tori Compact Vs. Algebraic Tori The l-dimensional algebraic torus is (C ) l with the same operation as the compact torus. A weight matrix A can also be used to describe a subgroup of GL n (C) that can be identified with (C ) l (just remove the requirement that t i = 1). The weight matrix (1) describes C as it is defined, i.e. matrices (t) with t 0. The weight matrix ( 2, 3) describes C as the subgroup {( ) } t t 3 : t C GL 2 (C). ( ) 2 3 The weight matrix describes (C 0 4 ) 2 as the subgroup {( ) } t t1 3 : (t 1, t 2 ) (C ) 2 GL 2 (C). t4 2 UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

54 Representations of Tori Compact Vs. Algebraic Tori Let A be an l n weight matrix. Let G denote the corresponding subgroup of GL n (C) identified with T l. Let G C denote the corresponding subgroup of GL n (C) identified with (C ) l. The subgroup G C is a kind of algebraic completion of G called the Zariski closure. We say that G C is the complexification of the Lie group G, as G is a maximal compact (closed and bounded) subgroup of G C. It is not hard to show (using the descriptions of the matrices) that a monomial in C[x 1,..., x n ] is G-invariant if and only if it is G C -invariant. Hence, C[x 1,..., x n ] G = C[x 1,..., x n ] G C. UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

55 Representations of Tori Compact Tori as Subgroups of GL n (R) We can also realize T l as a subgroup of GL n (R). For instance, S 1 can be identified with the group of rotations of the plane: {( ) } cos θ sin θ : θ R GL sin θ cos θ 2 (R). However, this is the same as S 1 GL 1 (C), identifying (x, y) R 2 with x + iy C. In general, any subgroup of GL n (R) that is a torus arises from a subgroup of GL m (C) for some m n/2. UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

56 Representations of Tori Real Invariants of Compact Tori If A is an l n weight matrix describing a subgroup of GL n (C), and hence a subgroup G of GL 2n (R), we can describe the point with coordinates (z 1,..., z n ) C n (x 1,..., x n, y 1,..., y n ) R 2 as above, z j = x j + iy j, or we can use where z j = x j iy j. (z 1,..., z n, z 1,..., z n ), The group operation in real coordinates is much easier to describe using these coordinates. UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

57 Representations of Tori Real Invariants of Compact Tori If A is an l n weight matrix describing a subgroup of GL n (C), and hence a subgroup G of GL 2n (R), a Hilbert basis for R[x 1,..., x n, y 1,..., y n ] G = R[z 1,..., z n, z 1,..., z n ] G is the same as a Hilbert basis of C[z 1,..., z n, w 1,..., w n ] H where H is the subgroup of GL 2n (C) corresponding to the weight matrix [A A]. UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

58 Representations of Tori Real Invariants of Compact Tori, Example ( ) The weight matrix A = describes a subgroup of GL (C) but also a subgroup G of GL 6 (R). To find a Hilbert basis for the G-invariants, we need only find a Hilbert basis for the invariants of the subgroup of GL 6 (C) associated to ( ) UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

59 Finding invariants using Gröbner bases Finding invariants using Gröbner bases UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

60 Finding invariants using Gröbner bases Algorithm for Torus Invariants This algorithm is from Sturmfels [5, Algorithm 1.4.5] Given an l n weight matrix A: Give C[t 1,..., t l, x 1,..., x n, y 1,..., y n ] a monomial order such that for any i, j, k, t i x j y k. For each column j of A, define q j = x j y j t a 1,j 1 t a 2,j 2 t a l,j l. If q j is not a polynomial (as some a i,j is negative), multiply by t a i,j i so that it is. Compute the reduced Gröbner basis for q 1,..., q n with respect to. The Hilbert basis of the invariants is the set of all x p 1 1 x n pn x p 1 1 x n pn y p 1 1 y n pn appears in the Gröbner basis. such that UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

61 Finding invariants using Gröbner bases Algorithm for Torus Invariants on Mathematica On Mathematica, we can compute this Gröbner basis with the command GroebnerBasis[ideal, variables, t-variables] where ideal is the list of the q j (in brackets {}), variables is the list of the all variables (t i s, then x j s, then y j s, all in brackets {}), and t-variables is the list of t i s (in brackets {}). UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

62 Finding invariants using Gröbner bases Algorithm for Torus Invariants on Mathematica, Example For the weight matrix ( ), we work in C[t 1, x 1, x 2, x 3, y 1, y 2, y 3 ]. We start with q 1 = x 1 y 1 t 2 1, q 2 = x 2 y 2 t 3 1, q 3 = x 3 y 3 t 5 1. But q 1 is not a polynomial, so we redefine q 1 = x 1 t 2 1 y 1. Hence, we enter: GroebnerBasis[{x1*t1^2-y1, x2-y2*t1^3, x3-y3*t1^5}, {t1,x1,x2,x3,y1,y2,y3}, {t1}] UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

63 Finding invariants using Gröbner bases Algorithm for Torus Invariants on Mathematica, Example ( ) : The output of is GroebnerBasis[{x1*t1^2-y1, x2-y2*t1^3, x3-y3*t1^5}, {t1,x1,x2,x3,y1,y2,y3}, {t1}] -x3^3 y2^5 + x2^5 y3^3, x1 x3 y2 - x2 y1 y3, -x3^2 y1 y2^4 + x1 x2^4 y3^2, -x3 y1^2 y2^3 + x1^2 x2^3 y3, x1^3 x2^2 - y1^3 y2^2, x1^4 x2 x3 - y1^4 y2 y3, x1^5 x3^2 - y1^5 y3^2 Hence, the Hilbert basis is {x 3 1 x 2 2, x 4 1 x 2 x 3, x 5 1 x 2 3 }. UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

64 Finding invariants using Gröbner bases Algorithm for Torus Invariants on Mathematica, Example ( ) For the weight matrix, we work in C[t 1,, t 2 x 1, x 2, x 3, x 4, y 1, y 2, y 3, y 4 ]. We start with q 1 = x 1 y 1 t 1 1, q 2 = x 2 y 2 t 2 2, q 3 = x 3 y 3 t 2 1t 3 2, q 4 = x 4 y 4 t 3 1t 4 2, and redefine q 1 = x 1 t 1 y 1 and q 2 = x 2 t 2 2 y 2. Hence, we enter: GroebnerBasis[{x1*t1 - y1, x2*t2^2 - y2, x3 - y3*t1^2*t2^3, x4 - y4*t1^3*t2^4}, {t1, t2, x1, x2, x3, x4, y1, y2, y3, y4}, {t1, t2}] UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

65 Finding invariants using Gröbner bases Algorithm for Torus Invariants on Mathematica, Example ( ) : The output of is GroebnerBasis[{x1*t1 - y1, x2*t2^2 - y2, x3 - y3*t1^2*t2^3, x4 - y4*t1^3*t2^4}, {t1, t2, x1, x2, x3, x4, y1, y2, y3, y4}, {t1, t2}] -x4^4 y2 y3^6 + x2 x3^6 y4^4, x1 x4^3 y3^4 - x3^4 y1 y4^3, -x4 y1 y2 y3^2 + x1 x2 x3^2 y4, x1^2 x2 x4^2 y3^2 - x3^2 y1^2 y2 y4^2, x1^3 x2^2 x4 - y1^3 y2^2 y4, x1^4 x2^3 x3^2 - y1^4 y2^3 y3^2 Hence, the Hilbert basis is {x 3 1 x 2 2 x 4, x 4 1 x 3 2 x 2 3 }. UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

66 Finding invariants using Gröbner bases Algorithm for Torus Invariants on Mathematica, Example For the weight matrix ( ), we use q 1 = x 1 t 1 y 1, q 2 = x 2 t 1 y 2, q 3 = x 3 y 3 t1, 2 q 4 = x 4 y 4 t1. 7 The input is GroebnerBasis[{ x1*t1^1 - y1, x2*t1^1 - y2, x3 - y3*t1^2, x4 - y4*t1^7}, {t1, x1, x2, x3, x4, y1, y2, y3, y4}, {t1}] UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

67 Finding invariants using Gröbner bases Algorithm for Torus Invariants on Mathematica, Example ( ) : The output is -x4^2 y3^7 + x3^7 y4^2, x2 x4 y3^3 - x3^3 y2 y4, -x4 y2 y3^4 + x2 x3^4 y4, x2^2 x3 - y2^2 y3, x2^3 x4 y3^2 - x3^2 y2^3 y4, x2^5 x4 y3 - x3 y2^5 y4, x2^7 x4 - y2^7 y4, -x2 y1 + x1 y2, x1 x4 y3^3 - x3^3 y1 y4, -x4 y1 y3^4 + x1 x3^4 y4, x1 x2 x3 - y1 y2 y3, x1 x2^2 x4 y3^2 - x3^2 y1 y2^2 y4, x1 x2^4 x4 y3 - x3 y1 y2^4 y4, x1 x2^6 x4 - y1 y2^6 y4, x1^2 x3 - y1^2 y3, x1^2 x2 x4 y3^2 - x3^2 y1^2 y2 y4, x1^2 x2^3 x4 y3 - x3 y1^2 y2^3 y4, x1^2 x2^5 x4 - y1^2 y2^5 y4, x1^3 x4 y3^2 - x3^2 y1^3 y4, x1^3 x2^2 x4 y3 - x3 y1^3 y2^2 y4, x1^3 x2^4 x4 - y1^3 y2^4 y4, x1^4 x2 x4 y3 - x3 y1^4 y2 y4, x1^4 x2^3 x4 - y1^4 y2^3 y4, x1^5 x4 y3 - x3 y1^5 y4, x1^5 x2^2 x4 - y1^5 y2^2 y4, x1^6 x2 x4 - y1^6 y2 y4, x1^7 x4 - y1^7 y4 Hence, the Hilbert basis is {x 2 2 x 3, x 7 2 x 4, x 1 x 2 x 3, x 1 x 6 2 x 4, x 2 1 x 3, x 2 1 x 5 2 x 4, x 3 1 x 4 2 x 4, x 4 1 x 3 2 x 4, x 5 1 x 2 2 x 4, x 6 1 x 2 x 4, x 7 1 x 4 }. UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

68 Finding invariants using Gröbner bases Other Kinds of Groups There are similar algorithms using Gröbner bases to compute Hilbert bases of invariants of finite groups, general compact Lie groups, etc. See Cox Little O Shea [2], Derksen and Kemper [3], and Sturmfels [5]. UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

69 Finding Relations Using Gröbner Bases Finding Relations Using Gröbner Bases UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

70 Finding Relations Using Gröbner Bases Algorithm for Relations This algorithm is from Cox Little O Shea [2, Proposition 7.4.3]. Given generators f 1,..., f r for a subalgebra of K[x 1,..., x n ] (e.g. a Hilbert basis): Give K[x 1,..., x n, y 1,..., y r ] a monomial order such that for each i, j, x i y j. Compute a Gröbner basis for the ideal I = f 1 y 1, f 2 y 2,..., f r y r. A Gröbner basis for the ideal of relations of f 1,..., f r is given by intersecting the result with K[y 1,..., y m ], i.e. removing the elements that involve the x i. UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

71 Finding Relations Using Gröbner Bases Algorithm for Relations on Mathematica On Mathematica, we can compute this Gröbner basis with the command: GroebnerBasis[ideal, variables, x-variables] where ideal is the list of f j y j, j = 1,..., r (in brackets {}), variables is the list of the all variables (x j s, then y j s, all in brackets {}), and x-variables is the list of x j s (in brackets {}). UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

72 Finding Relations Using Gröbner Bases Algorithm for Relations on Mathematica, Example Recall that the weight matrix ( ) had Hilbert basis {x 3 1 x 2 2, x 4 1 x 2 x 3, x 5 1 x 2 3 }. We work in C[x 1, x 2, x 3, y 1, y 2, y 3 ] and enter: GroebnerBasis[{ x1^3*x2^2-y1, x1^4*x2*x3-y2, x1^5*x3^2-y3}, {x1,x2,x3,y1,y2,y3}, {x1,x2,x3}] The output is: -y2^2 + y1 y3 So the ideal of relations is f 1 f 3 f 2 2. UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

73 Finding Relations Using Gröbner Bases Algorithm for Relations on Mathematica, Example Recall that the weight matrix ( ) had Hilbert basis {x 3 1 x 2 2 x 4, x 4 1 x 3 2 x 2 3 }. We work in C[x 1, x 2, x 3, x 4, y 1, y 2 ] and enter: GroebnerBasis[{ x1^3*x2^2^2*x4-y1, x1^4*x2^3*x3^2-y2}, {x1,x2,x3,x4,y1,y2}, {x1,x2,x3,x4}] The output is empty, so there are no relations. UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

74 Finding Relations Using Gröbner Bases Algorithm for Relations on Mathematica, Example Recall that the weight matrix ( ) had Hilbert basis {x 2 2 x 3, x 7 2 x 4, x 1 x 2 x 3, x 1 x 6 2 x 4, x 2 1 x 3, x 2 1 x 5 2 x 4, x 3 1 x 4 2 x 4, x 4 1 x 3 2 x 4, x 5 1 x 2 2 x 4, x 6 1 x 2 x 4, x 7 1 x 4 }. We work in C[x 1, x 2, x 3, x 4, y 1, y 2, y 3, y 4, y 5, y 6, y 7, y 8, y 9, y 10, y 11 ] and enter: GroebnerBasis[{ x2^2*x3-y1, x2^7*x4-y2, x1*x2*x3-y3, x1*x2^6*x4-y4, x1^2*x3-y5, x1^2*x2^5*x4-y6, x1^3*x2^4*x4-y7, x1^4*x2^3*x4-y8, x1^5*x2^2*x4-y9, x1^6*x2*x4-y10, x1^7*x4-y11 }, {x1,x2,x3,x4,y1,y2,y3,y4,y5,y6,y7, y8,y9,y10,y11}, {x1,x2,x3,x4}] UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

75 Finding Relations Using Gröbner Bases Algorithm for Relations on Mathematica, Example The output is -y10^2 + y11 y9, y11 y8 - y10 y9, y10 y8 - y9^2, y11 y7 - y9^2, y10 y7 - y8 y9, -y8^2 + y7 y9, y11 y6 - y8 y9, y10 y6 - y8^2, -y7 y8 + y6 y9, -y7^2 + y6 y8, y11 y4 - y8^2, y10 y4 - y7 y8, -y7^2 + y4 y9, -y6 y7 + y4 y8, -y6^2 + y4 y7, y11 y3 - y10 y5, y10 y3 - y5 y9, -y5 y8 + y3 y9, -y5 y7 + y3 y8, -y5 y6 + y3 y7, -y4 y5 + y3 y6, y11 y2 - y7 y8, y10 y2 - y7^2, -y6 y7 + y2 y9, -y6^2 + y2 y8, -y4 y6 + y2 y7, -y4^2 + y2 y6, -y3 y4 + y2 y5, y1 y11 - y5 y9, y1 y10 - y5 y8, -y5 y7 + y1 y9, -y5 y6 + y1 y8, -y4 y5 + y1 y7, -y3 y4 + y1 y6, -y3^2 + y1 y5, -y2 y3 + y1 y4 UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

76 Finding Relations Using Gröbner Bases Algorithm for Relations on Mathematica, Example Hence, there are 36 relations. f f 11 f 9, f 11 f 8 f 10 f 9, f 10 f 8 f9 2, f 11 f 7 f9 2, f 10 f 7 f 8 f 9, f8 2 + f 7 f 9, f 11 f 6 f 8 f 9, f 10 f 6 f8 2, f 7 f 8 + f 6 f 9, f7 2 + f 6 f 8, f 11 f 4 f8 2, f 10 f 4 f 7 f 8, f7 2 + f 4 f 9, f 6 f 7 + f 4 f 8, f6 2 + f 4 f 7, f 11 f 3 f 10 f 5, f 10 f 3 f 5 f 9, f 5 f 8 + f 3 f 9, f 5 f 7 + f 3 f 8, f 5 f 6 + f 3 f 7, f 4 f 5 + f 3 f 6, f 11 f 2 f 7 f 8, f 10 f 2 f7 2, f 6 f 7 + f 2 f 9, f6 2 + f 2 f 8, f 4 f 6 + f 2 f 7, f4 2 + f 2 f 6, f 3 f 4 + f 2 f 5, f 1 f 11 f 5 f 9, f 1 f 10 f 5 f 8, f 5 f 7 + f 1 f 9, f 5 f 6 + f 1 f 8, f 4 f 5 + f 1 f 7, f 3 f 4 + f 1 f 6, f3 2 + f 1 f 5, f 2 f 3 + f 1 f 4 UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77

77 References: Finding Relations Using Gröbner Bases Thank you! Jacek Bochnak, Michel Coste, and Marie-Françoise Roy, Real algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 36, Springer-Verlag, Berlin, 1998, Translated from the 1987 French original, Revised by the authors. MR (2000a:14067) David A. Cox, John Little, and Donal O Shea, Ideals, varieties, and algorithms, fourth ed., Undergraduate Texts in Mathematics, Springer, Cham, 2015, An introduction to computational algebraic geometry and commutative algebra. Harm Derksen and Gregor Kemper, Computational invariant theory, Invariant Theory and Algebraic Transformation Groups, I, Springer-Verlag, Berlin, 2002, Encyclopaedia of Mathematical Sciences, 130. V. L. Popov and È. B. Vinberg, Invariant theory, Algebraic geometry. IV, Encyclopaedia of Mathematical Sciences, vol. 55, Linear algebraic groups. Invariant theory, A translation of ıt Algebraic geometry. 4 (Russian), Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1989, Translation edited by A. N. Parshin and I. R. Shafarevich. Bernd Sturmfels, Algorithms in invariant theory, second ed., Texts and Monographs in Symbolic Computation, SpringerWienNewYork, Vienna, UFRJ; H.-C. Herbig, C. Seaton Computational methods symplectic quotients January th, / 77