Orthogonal matrix invariants

The orthogonal group acts on the space of several $n\times n$ matrices by simultaneous conjugation. For an infinite field of characteristic different from two, relations between generators for the algebra of invariants are described. As an application, the maximal degree of elements of a minimal system of generators is described with deviation $3$. This note contains concise but precise description of the results. All proofs can be found in arXiv: 0902.4266 and arXiv: 1011.5201.


Introduction
All vector spaces, algebras, and modules are over an infinite field F. By an algebra we always mean an associative algebra.
Let a linear group G be a subgroup of GL(n) and be d-tuple of n × n matrices over F. The group G acts on V by the diagonal conjugation, i.e., g • (A 1 , . . ., A d ) = (gA 1 g −1 , . . ., gA d g −1 ), where g ∈ G and A 1 , . . ., A d ∈ F n×n .The coordinate ring of V (i.e. the ring of polynomial functions f : V → F) is the ring of polynomials where x ij (k) stands for the coordinate function on V that takes (A 1 , . . ., A d ) ∈ V to the (i, j) th entry of the matrix A k .Denote by The action of G on V induces the action on F[V ] as follows: The algebra of matrix G-invariants is If G is a classical linear group, i.e., G belongs to the list GL(n), O(n), Sp(n), SO(n), SL(n), then a generating set for the algebra F[V ] G is known for an arbitrary characteristic of F (see [2], [12], [7]), where we assume that char F = 2 in the case of O(n) and SO(n).
Theorem 1.1.If char F = 2, then the algebra of matrix invariants where B ranges over all monomials in X 1 , . . ., X d , X T 1 , . . ., X T d .Moreover, we can assume that B is primitive, i.e., is not equal to a power of a shorter monomial.Remark 1.2.In the case of a characteristic zero field it is enough to take traces instead of σ t , 1 ≤ t ≤ n, in the formulation of Theorem 1.1.
In characteristic zero case Procesi [10] described relations between generators for F[V ] G for G ∈ {GL(n), O(n), Sp(n)}.Zubkov [11] described relations for matrix GL(n)-invariants over a field of arbitrary characteristic.We have described relations for matrix O(n)-invariants (see Theorem 2.1) in the case of characteristic different from two.The proofs of Theorem 2.1 and Lemma 3.1 will be given in a separate paper.

Relations
For a vector t = (t 1 , . . ., t u ) ∈ N u we write #t = u, where N stands for the set of non-negative integers.In this paper we use the following notions: • the monoid M (without unity) freely generated by letters x 1 , . . ., x d , x T 1 , . . ., x T d , the vector space M F with the basis M, and N ⊂ M the subset of primitive elements, where the notion of a primitive element is defined as above; • the involution T : M F → M F defined by x T T = x for a letter x and (a where y 1 , . . ., y p , z 1 , . . ., z p are letters; • M σ , the ring with unity of (commutative) polynomials over F freely generated by the "symbolic" elements σ t (α), where t > 0 and α ∈ M F ; • N σ , a ring with unity of (commutative) polynomials over F freely generated by the "symbolic" elements σ t (α), where t > 0 and α ∈ N ranges over ∼-equivalence classes; note that N σ ≃ M σ /L, where the ideal L is described in Lemma 3.1 (see below).
The following result was known in characteristic zero case.
Corollary 2.2.In case G = O(n) there are no non-zero free relations.
3 The definition of σ t,r In this section we assume that A is a commutative unitary algebra over the field F and all matrices are considered over A.
Let us recall some formulas.In what follows A, A 1 , . . ., A p stand for n × n matrices and 1 ≤ t ≤ n.Amitsur's formula states [1]: where the sum ranges over all pairwise different primitive cycles γ 1 , . . ., γ q in letters A 1 , . . ., A p and positive integers j 1 , . . ., j q with q i=1 j i deg γ i = t.Denote the right hand side of (3) by F t,p (A 1 , . . ., A p ).As an example, Note that for a ∈ A we have σ t (aA) = a t σ t (A). ( For l ≥ 2 we have the following well-known formula: where we assume that σ i (A) = 0 for i > n.Denote the right hand side of ( 6) by P t,l (A).In ( 6) coefficients b (t,l) i1,...,i rl ∈ Z do not depend on A and n.If we take a diagonal matrix A = diag(a 1 , . . ., a n ), then σ t (A l ) is a symmetric polynomial in a 1 , . . ., a n and σ i (A) is the i th elementary symmetric polynomial in a 1 , . . ., a n , where 1 ≤ i ≤ n.Thus the coefficients b (t,l) i1,...,i tl with tl ≤ n can easily be found.As an example, tr Lemma 3.1.We have N σ ≃ M σ /L for the ideal L generated by where p > 1, α, α 1 , . . ., α p ∈ M F , a ∈ F, t > 0, and l > 1.

We say that
The head of the path α is α ′ = α ′ 1 and the tail is Assume that Q is a mixed quiver.Denote by M(Q) the set of all closed paths in Q and denote by N (Q) ⊂ M(Q) the subset of primitive paths.Given a path α in Q, we define the path α T and introduce ∼-equivalence on M(Q) in the same way as in Section 2.Moreover, we define M F (Q), M σ (Q), and N σ (Q) in the same way as M F , M σ , and N σ have been defined in Section 2.
The decomposition formula from [5] implies that for n × n matrices A where DP r,r (A 1 , A 2 , A 3 ) stands for the determinant-pfaffian (see [6]) and σ t,r (A 1 , A 2 , A 3 ) is defined as the result of the substitution a i → A i , a T i → A T i in σ t,r (a 1 , a 2 , a 3 ).Thus DP r,r relates to σ t,r in the same way as the determinant relates to σ t .

Application
Given an N-graded algebra A, denote by A + the subalgebra generated by elements of A of positive degree.It is easy to see that a set {a i } ⊆ A is a minimal (by inclusion) homogeneous system of generators (m.h.s.g.) for A if and only if {a i } is a basis for A = A/(A + ) 2 and {a i } are homogeneous.Let us recall that an element a ∈ A is called decomposable if it belongs to the ideal (A + ) 2 .Therefore the least upper bound for the degrees of elements of a m.h.s.g. for F[V ] O(n) is equal to the maximal degree of indecomposable invariants and we denote it by D max .
As an application of Theorem 2.1 we obtained the following result in [8].• If char F = 2, 3, then D max = 6.
Moreover, in [9] we described a m.h.s.g. for orthogonal invariants of d-tuples of 3 × 3 skewsymmetric matrices.As about matrix GL(n)-invariants in case n = 3, its minimal system of generators was explicitly calculated in [3] and [4].