The Waring rank of the Vandermonde determinant

The Waring rank of the Vandermonde determinant Alexander Woo (U. Idaho) joint work with Zach Teitler(Boise State) SIAM Conference on Applied Algebraic Geometry, August 3, 2014

Waring rank Given a polynomial f S = C[x 1,..., x n ], homogeneous of degree d, the Waring rank of f is the least number r of linear forms L 1,..., L r such that f = L d 1 + + L d r. This can also be thought of as the symmetric rank of a symmetric tensor. The Waring rank of a generic polynomial of degree d in n variables is known by a theorem of Alexander and Hirschowitz (1995). The rank of a specific polynomial can be both less than and greater than this generic rank.

Apolarity Let T = C[ 1,..., n ] be the ring of partial differentiation operators acting on S. Given a polynomial f S, define f = { T f = 0}. Note f T is a 0-dimensional ideal. The well-known Apolarity Lemma states that f = L d 1 + + L d r if and only if I ({L 1,..., L r }) f.

Ranestad Schreyer bound A homogeneous ideal I is generated in degree δ if there exist polynomials g 1,..., g k I, all of degree δ, such that I = g 1,..., g k. The Apolarity Lemma says that the Waring rank of f is a property of the ideal f. Ranestad and Schreyer show that we can easily read off the following lower bound: r(f ) dim C T /f. δ where δ is any integer such that f is generated in degree δ.

The Vandermonde determinant The Vandermonde determinant V n is the determinant of the matrix 1 1 1 1 x 1 x 2 x n 1 x n..... x1 n 2 x2 n 2 xn 1 n 2 xn n 2 x1 n 1 x2 n 1 xn 1 n 1 xn n 1 Also, V n = 1 i<j n (x j x i ).

Apolarity for the Vandermonde It is classically known that Vn = e 1 ( 1,..., n ),..., e n ( 1,..., n ), where e i is the i-th elementary symmetric function i e i = x jk. 1 j 1 < <j i n k=1 Furthermore, Vn is a complete intersection, with generators in degrees 1, 2,..., n.

Apolarity for the Vandermonde It is classically known that V n = e 1 ( 1,..., n ),..., e n ( 1,..., n ), where e i is the i-th elementary symmetric function e i = 1 j 1 < <j i n k=1 i x jk. Furthermore, Vn is a complete intersection, with generators in degrees 1, 2,..., n. Therefore, the Ranestad Schreyer bound implies r(v n ) n! n = (n 1)!.

Vandermonde and anti-symmetrization The Vandermonde V n is (up to a multiple) the unique polynomial of degree D := ( n 2) on which any permutation σ Sn acts by multiplying by sgn(σ). This means if f is any polynomial of degree D, then alt(f ) := 1 sgn(σ)σ(f ) = kv n n! σ S n for some k C (possibly k = 0). Since σ(l D ) = σ(l) D, and alt(l D ) has n! terms, this shows r(v n ) n!.

Explicit Waring decomposition for Vandermonde To do better, one picks a linear form L (with alt(l D ) 0) so that σ(l) is a multiple of L for all σ in some large subgroup C S n. We choose L = ωx 1 + ω 2 x 2 + + ω n 1 x n 1 + x n, where ω is a primitive n-th root of unity.

The rank of the Vandermonde If σ is the n-cycle σ = (12 n), then σ(l) = ωl, and σ m (L) = ω m L. We can check explicitly that alt(l D ) 0 (e.g. by checking a particular coefficient). This means σ(l) and τ(l) will generate the same term (up to a constant) in alt(l D ) anytime σ and τ are in the same coset of C = (12 n), so r(v n ) = (n 1)!. An explicit decomposition is (x i x j ) = i<j 1 (n 1)! σ(n)=n ( n n 2) sgn(σ) ω j x j. j=1

Other representations of S n Waring rank can be defined for any homogeneous subspace W S d = C[x 1,..., x n ] d. Define r(w ) to be the smallest number of linear forms L 1,..., L r such that Span(L d 1,..., Ld r ) W. The irreducible representations of S n are parameterized by partitions λ of n. Each irreducible representation of S n has a distinguished occurrence G λ of lowest degree in S, so one can ask for the Waring rank r(g λ ).

Apolarity for other representations Let H λ = T /G λ. Then H λ is the cohomology ring of the Springer fiber, which is equal to the coordinate ring of the scheme theoretic intersection of a nilpotent orbit with the variety of diagonal matrices. The dimension (and graded S n character) of H λ is known, as are various generating sets for Gλ. None of the known generating sets are minimal in general. Can looking at Waring rank give some information on the problem of finding minimal generating set, at least by giving a lower bound on the generating degree?

(Complex) reflection groups A complex reflection t is an orthogonal matrix which fixes a hyperplane H t (so the eigenvalue 1 has geometric multiplicity n 1) and one eigenvalue which is a primitive k t -th root of unity for some k t. If k t = 2, t is a real reflection. A complex reflection group W is a finite subgroup of SO(V ) which is generated by complex reflections. If all reflections are real W is a real reflection group or simply a reflection group. The group S n is a reflection group acting on a vector space of dimension n. The transposition (ij) fixes the hyperplane defined by x i x j = 0, and sends e i e j to (e i e j ).

Classification of real reflection groups Irreducible real reflection groups are classified by the Dynkin classification: A n 1 is S n. B n = C n is the hyperoctahedral group Z/2Z S n of signed permutations. D n is an index 2 subgroup of B n. E 6, E 7, E 8, F 4, and G 2 are exceptional Weyl groups. H 3 (icosahedral group), and H 4 (120-cell) are non-crystallographic. I 2 (m) are the dihedral groups, also non-crystallographic (except m = 2, 3, 4, 6).

Classification of complex reflection groups Irreducible complex reflection groups were classified by Shepard and Todd in the 1950s. There is one family G(m, p, n) (with p dividing n), including types A, B, and D, cyclic and dihedral groups. There are also 34 exceptional groups. Given two reflection groups, their direct product is also a reflection group.

Skew Invariants for a Reflection Group The generalization of the Vandermonde determinant is the fundamental skew invariant f W. If L t denotes the linear form vanishing on the reflecting hyperplane H t, then f W = L kt 1 t. t

The covariant ring of a complex reflection group Let I W be the ideal in T = Sym (V ) generated by all non-constant W -invariant polynomials. A classical theorem of Steinberg states the following for any complex reflection group: f W = I W. I W is a complete intersection ideal (minimally generated by n = dim V elements). dim C (T /I W ) = W

The degrees of a complex reflection group The degrees of the minimal generators of I W are known as the degrees of W and customarily denoted d 1,..., d n. In the case W = S n, the minimal generators of I W are e 1,..., e n, so the degrees are 1,..., n. The product of the degrees is W. The sum of the degrees is the degree of f W. The Ranestad Schreyer bound says r(f W ) d 1 d n 1 = W /d n.

Anti-symmetrization for a reflection group The generalization of the sign representation of S n is the 1-dimensional determinant representation of W, where σ W acts by σ(f ) = det(σ)f. Any element of S = Sym (V ) on which W acts as the determinant representation is divisible by f W, so alt(f ) = (det w)w(f ) = kf W w W for some k (possibly 0) whenever deg(f ) = deg(f W ). This gives an upper bound r(f W ) W.

Coxeter (or regular) elements A linear form in V is regular if it does not lie on a reflecting hyperplane. It turns out alt(l D ) 0 precisely when L is regular. The analogue of the cycle (1 n) is a regular element; this is an element w W which has a regular linear form as an eigenvector. Here we want a regular element of order d n. Real reflection groups always have a regular element of order d n, known as a Coxeter element. A large class but not all of non-real complex reflection groups have a regular element of order d n.

General theorem If W has a regular element c of order m, let L be a regular eigenvector for c. Then alt(l D ) actually only has W /m distinct terms. So if d n is the order of a regular element, then we have an explicit Waring decomposition with W /d n terms. With the Ranestad Schreyer bound, this shows the following:

General theorem If W has a regular element c of order m, let L be a regular eigenvector for c. Then alt(l D ) actually only has W /m distinct terms. So if d n is the order of a regular element, then we have an explicit Waring decomposition with W /d n terms. With the Ranestad Schreyer bound, this shows the following: Theorem Let W be a finite complex reflection group acting on C n such that d n is the order of a regular element. Then r(f W ) = W /d n.

Example for B n For B n we get n x i (x i x j )(x i + x j ) = i=1 where i<j L = 1 2 n 1 (n 1)! n e 2πi(j 1)/2n x j j=1 σ B n/c sgn(σ)(σ L) n2, and C is generated by the element 23 n1 (one-line notation).

Monomials A monomial x a 1 1 x n an is the fundamental skew-invariant for the group Z/(a 1 + 1)Z Z/(a n + 1)Z. However, the largest degree is not the order of a regular element unless a 1 = = a n. In this case, our result was observed by Ranestad and Schreyer. The Waring rank and explicit decompositions are known for all monomials; this may provide a hint on how to remove the hypothesis that d n is the order of a regular element.

The End Thank you for your attention!

References This talk is based on the paper Apolarity and reflection groups, ArXiv: 1304.7202. The Ranestad Schreyer paper is On the rank of a symmetric form, J. Algebra 346 (2011), 340 342. For more on complex reflection groups, see the book by Gus Lehrer and Donald Taylor, Unitary reflection groups, or the classic on real reflection groups by Jim Humphreys, Reflection groups and Coxeter groups. For recent work on the apolar ideals for other representations of S n, see the paper by Biagioli, Faridi, and Rosas, The defining ideals of conjugacy classes of nilpotent matrices and a conjecture of Weyman, IMRN 2008, Art. ID rnn117.