THE INVARIANT THEORY OF K−1[X1;X2;:::;XN ] UNDER PERMUTATION REPRESENTATIONS BY J. CAMERON ATKINS A Thesis Submitted to the Graduate Faculty of WAKE FOREST UNIVERSITY GRADUATE SCHOOL OF ARTS AND SCIENCES in Partial Fulfillment of the Requirements for the Degree of MASTER OF ARTS Mathematics May 2012 Winston-Salem, North Carolina Approved By: Ellen Kirkman, Ph.D., Advisor Sarah K. Mason, Ph.D., Chair W. Frank Moore, Ph.D. Acknowledgments I would like to thank my parents for being so patient with me through my adoles- cents. When I decided to pursue a path in mathematics, they supported me all the way, and they continue to support me as I head to the University of South Carolina. My siblings Jared and Whitney treated me as the younger of the three, and over the past years we have grown to truly love each other. I feel lucky to have the relationship we share and encouragement they give me. I would also like to thank Melissa Bechard. She and I began studying mathematics our freshmen year at James Madison Univer- sity. We struggle and we fight through problems together, and if nothing else, I have fun exploring math with her. Lastly, I would like to thank my advisor Dr. Kirkman. Out of everyone she is the most deserving of my praise. As a professor, I have never learned more from one individual. Her rigor from the classroom reappeared in her guidance as an advisor, and I am a better mathematician for it. ii Table of Contents Acknowledgments . ii Abstract . iv Chapter 1 Introduction . 1 1.1 Groups and Rings . 1 1.2 Group Representation . 2 1.3 Invariant Theory of k[x1; x2 : : : ; xn] Under Permutation Matrices . 2 1.4 The Non-Commutative Ring k−1[x1; x2; ··· ; xn] ............ 5 Chapter 2 Orbit Sums and the Symmetric Group. 7 2.1 Orbit Sums . 7 2.2 Symmetric Group . 9 Chapter 3 G¨obel's Bound . 19 3.1 Special Monomials . 19 3.2 G¨obel's Bound . 21 Chapter 4 The Alternating Group. 24 4.1 Antisymmetric Invariants . 24 4.2 RAn Generators . 28 Chapter 5 Examples With G¨obel's Bound with 4 indeterminates. 33 5.1 Alternating Z2 Subgroup . 33 5.2 Alternating Klein IV Subgroup . 36 5.3 Non-Alternating Klein IV Subgroup . 38 Bibliography . 42 Appendix A 5.1 . 43 Appendix B 5.2 . 46 Vita............................................................................ 48 iii Abstract Let k be a field of characteristic zero, and let R = k−1[x1; ··· ; xn] denote the skew- polynomial ring with xjxi = −xixj for i 6= j. The symmetric group Sn acts on R as permutations of the fxig. Given a subgroup G of permutations in Sn we consider the problem of finding algebra generators for RG, the subring of invariants under G G. We find generators for R when G is the full symmetric group Sn and when G is the alternating group An. We describe an algorithm that produces generators of G R for a general subgroup G of Sn. The algorithm produces a an upper bound on the degrees of algebra generators of RG, generalizing the G¨obel bound for invariants under permutation actions on k[x1; ··· ; xn]. iv Chapter 1: Introduction 1.1 Groups and Rings Let G be a group and S be a set. We define a left group action on S by G to be G × S ! S, where (g; s) = gs, and if e is the identity of G the action satisfies es = s and g(hs) = (gh)s for all g; h 2 G and for all s 2 S. We call the set S a G-set. The orbit of element s 2 S under a group G is the set of elements t in S such that there exists some g 2 G with gs = t. We denote the orbit of an element s 2 S under a group G to be OrbG(s). A stabilizer of element s 2 S is an element of g 2 G such that gs = s. The set of all stabilizers of an element s 2 S from G forms a subgroup of G denoted StabG(s). We let Sn denote the symmetric group on n elements and An be the alternating group. A ring is a set R with two binary operations + and · which are called addition and muliplication respectively. The ring under addition forms an abelian group. The ring under multiplication satisfies the left and right distribution laws, i.e. a(b + c) = ab + ac and (a + b)c = ac + bc, as well as being associative, i.e. a(bc) = (ab)c.A field is a commutative ring with unity where every non-zero element is a unit. The characteristic of a field, k, is the smallest positive integer n such that 8r 2 k, nr = 0. If no positive integer exists we say the ring has characteristic zero. Throughout this thesis k is always a field of characteristic zero. We can construct polynomials with n-indeterminates with coefficients from a field k denoted k[x1; x2; : : : ; xn] or R. Let I I = (i1; i2; : : : ; in) be a vector of length n of non-negative integers, and let X denote i1 i2 in the monomial x1 x2 ··· xn . 1 1.2 Group Representation A linear representation of a group G is a homomorphism ' : G ! Gl(n; k), where k is any field, and Gl(n; k) is the set of n × n invertible matrices with entries from k. A representation, ', is called faithful if the kernel of ' is only the identity of G. The −1 0 example ' : ! Gl(2; ), with '(1) = defines a representation fo , Z4 R 0 1 Z4 where '(j) = '(1 + 1 + ··· + 1) (one is added to itself j times). This representation is not fiathful to since ker(') = f0; 2g. On the other hand, ρ : Z3 ! Gl(2; C) 2πi e 3 0 where ρ(1) = defines a faithful representation since ker(ρ) = f0g.A 0 1 permutation matrix is denoted as Mσ, and is the matrix where the entries mi;j = 1 if σ(j) = i and mi;j =0 otherwise. For example consider S4 and the permutation σ = (1; 2; 3). Then the permutation matrix is 0 0 0 1 0 1 B 1 0 0 0 C Mσ = B C. @ 0 1 0 0 A 0 0 0 1 The representation of the symmetric group Sn with φ : Sn ! Gl(n; R) where φ(σ) = Mσ is called a permutation representation of Sn. This representation is faithful. In this thesis we explore the permuation representation of the symmetric group as well as the subgroups of the symmetric group under the same representation. 1.3 Invariant Theory of k[x1; x2 : : : ; xn] Under Permutation Ma- trices For any subgroup G of Sn, represented as permutation matrices, we define a G-action on the set of monomials. If σ is a permutation of f1; 2; : : : ; ng then we define i −1 i −1 i −1 i1 i2 in i1 i2 in σ (1) σ (2) σ (n) σ(x1 x2 ··· xn ) = xσ(1)xσ(2) ··· xσ(n) = x1 x2 ··· xn : 2 This action is extended to R = k[x1; ··· ; xn] by X I X I σ aI X = aI σ(X ): I I Given any subgroup G of Sn and an element f 2 R, we say f is invariant under G if 8g 2 G, g(f) = f. The set of elements of R that are invariant under G forms a subring of R that we denote by RG. This thesis concerns finding algebra generators G G for the invariant subring R , i.e. finding elements f1; ··· ; fm of R such that any invariant element f 2 RG can be written as sum of the form X i1 i2 im f = aI (f1 f2 ··· fm ) for aI 2 k: I=(i1;i2;:::;im) One way to produce invariants is to form orbit sums. For any monomial XI , let I I OG(X ) denote the sum of the elements in the G-orbit of X , namely I X I OG(X ) = g(X ): I OrbG(X ) Any orbit sum will be an invariant in RG. Moreover, it can be shown that any element of RG is a linear combination of orbit sums [2]. When G is the full symmetric group Sn an orbit sum can be represented by its leading term under the lexicographic order with x1 > x2 > ··· > xn; the exponent sequence of that leading term will be weakly decreasing and hence form a partition into n parts of the sum of the exponents. Hence, orbit sums under Sn correspond with partitions into n parts. When G = Sn Gauss proved that the n elementary symmetric polynomials are a generating set for RSn , where the elementary symmetric polynomials and the corresponding partitions are: e1(x1; : : : ; xn) = OSn (x1) = x1 + x2 + ··· + xn $ (1; 0; 0; ··· 0); X e2(x1; : : : ; xn) = OSn (x1x2) = xixj $ (1; 1; 0; ··· 0); i6=j 3 X e3(x1; : : : ; xn) = OSn (x1x2x3) = xixjxk $ (1; 1; 1; 0; ··· 0); i;j;k distinct . en(x1; : : : ; xn) = OSn (x1x2 ··· xn) = x1x2 ··· xn $ (1; 1; 1; ··· 1): Power sums form another algebra generating set for the symmetric functions RSn , where for 1 ≤ i ≤ n the n power sums are defined by i i i i pi = OSn (x1) = x1 + x2 + ··· + xn $ (i; 0; 0; ··· 0): Newton proved the Newton Symmetric Formulas, which are relations between the power sums and the elementary symmetric polynomials, and Waring gave a formula expressing the power sums in terms of the elementary symmetric polynomials [2].