Algorithms in Invariant Theory by Bernd Sturmfels Ch. 3.1: The Straightening Algorithm Summary Lecturer: Katherine Harris; Note-taker: Jane Coons Let X = (xij) be an n × d matrix whose entries are indeterminates, and let C[xij] be the corresponding polynomial ring in nd variables. Since we can think of X as a set of n row vectors in Cd, X can be used to represent n points in Pd−1. Definition. Let Λ(n; d) = f[λ1; : : : ; λd] j 1 ≤ λ1 < ··· < λd ≤ n]g. An element of Λ(n; d) n is called a bracket. Denotes by C[Λ(n; d)] the polynomial ring in d variables generated by the elements of Λ(n; d). Definition. Define φn;d : C[Λ(n; d)] ! C[xij] called the generic coordinization by φn;d(λ) = det(xλi;j)1≤i;j≤d. Under the generic coordinization, the bracket [λ] maps to the d × d subde- terminant of X with rows indexed by λ1; : : : ; λd. Example. Let d = 2, n = 3, and [λ] = [1; 3]. Then φ3;2([λ]) = x11x32 − x12x31: Notice that φn;d is a homomorphism. Also note that im(φn;d) is the subring of C[xij] generated by the d × d minors of X. We call this the bracket ring, Bn;d. The map φn;d is generally not injective. Let In;d = ker(φn;d). Then In;d is the ideal of syzygies among the d × d minors of X. Example. If d = 2 and n = 3, then I4;2 = h[12][34] − [13][24] + [14][23]i. ∼ Note that Bn;d = C[Λ(n; d)]=In;d. Definition. The projective variety defined by In;d is called the Grassmann variety whose points correspond to d-dimensional vector spaces of Cn. We wish to find a Gröbner basis for In;d. Definition. The complement of the bracket [λ] is [λ∗], the unique (n − d)-tuple such that [λ] [ [λ∗] = [n]. ∗ Definition. The sign of ([λ]; [λ ]) is sgnπ where π is the permutation that maps λi to i for ∗ i = 1; : : : ; d and λj to j + d for j = 1; : : : ; n − d. Definition. Let s 2 [d], α 2 Λ(n; s − 1), β 2 Λ(n; d + 1) and γ 2 Λ(n; d − s). Then the van der Waerden syzygy is defined by X ∗ [[αβγ_ ]] = sgn([τ]; [τ ])[α ; : : : ; α ; β ∗ ; : : : β ∗ ][β ; : : : ; β γ ; : : : ; γ ] 1 s−1 τ1 τd+1−s τ1 τs 1 d−s [τ]2Λ(d+1;s) Example. Let d = 3, n = 6 and s = 2. Let α = [1], β = [2356], γ = [4]. Then the van der Waerden syzygy is [[12_3_5_64]]_ = [156][234] + [136][245] − [135][245] − [126][345] + [125][346] + [123][456]: _ Definition. The van der Waerden syzygy [[αβγ]] is a straightening syzygy if αs−1 < βs + 1 and βs < γ1. Let Sn;d be the set of straightening syzygies. We order the elements of Λ(n; d) lexicographically. This induces a graded reverse lexico- graphic order on the elements of C[Λ(n; d)], which we will call tableau order. It is canonical to write elements in C[Λ(n; d)] as tableaux. Given [λ1];:::; [λk] 2 Λ(n; d) with [λ1] < ··· < [λk] in tableau order, we write the monomial T = [λ1] < ··· < [λk] as 2[λ1]3 . T = 4 . 5 [λk] Definition. A tableau T is standard if its columns are sorted. If not, it is nonstandard. Example. Let T1 = [123][124]. When we represent T1 as a tableau, it becomes 1 2 3 T = ; 1 1 2 4 whose columns are sorted. So T1 is standard. Let T2 = [145][234]. When we represent T2 as a tableau, it becomes 1 4 5 T = : 2 2 3 4 The second column in T2 is unsorted, so T2 is nonstandard. Theorem. The set of straightening syzygies, Sn;d is a Gröbner basis for In;d with respect to the tableau order. A tableau T is standard if and only if T 62 LT(In;d). Corollary. The standard tableau form a C-vector space basis for Bn;d. This corollary is known as the straightening law. Due to the straightening law, we can do normal form reductions of elements of Bn;d modulo In;d. This process is called the Straight- ening Algorithm. Example. Let n = 6 and d = 3. Let 21 4 53 T = 41 5 65 : 2 3 4 Note that T is not standard. We can apply the Straightening Algorithm to write a normal form reduction of T in terms of standard tableau, T = [123][145][456] − [124][145][3456] + [134][145][256] modulo I6;3:.