[Paper Review] Some schemes related to the commuting variety

Allen Knutson|arXiv (Cornell University)|Jun 18, 2003

Coding theory and cryptography2 references4 citations

TL;DR

This paper introduces the diagonal commutator scheme, defined as pairs of $n \times n$ matrices whose commutator is diagonal, proving it is a reduced complete intersection with the commuting variety as one component. Using a Gröbner degeneration via a one-parameter subgroup, the authors show the special fiber decomposes into $n!$ components indexed by permutations, each corresponding to a matrix Schubert variety, and conjecture the scheme has exactly two components for reductive Lie algebras.

ABSTRACT

The_commuting variety_ is the pairs of NxN matrices (X,Y) such that XY = YX. We introduce the_diagonal commutator scheme_, {(X,Y) : XY-YX is diagonal}, which we prove to be a reduced complete intersection, one component of which is the commuting variety. (We conjecture there to be only one other component.) The diagonal commutator scheme has a flat degeneration to the scheme {(X,Y) : XY lower triangular, YX upper triangular}, which is again a reduced complete intersection, this time with n! components (one for each permutation). The degrees of these components give interesting invariants of permutations.

Motivation & Objective

• To study the structure of the commuting variety in $\mathfrak{gl}_n$ by introducing a related scheme with fewer defining equations.
• To prove that the diagonal commutator scheme is a reduced complete intersection, one component of which is the commuting variety.
• To analyze the flat degeneration of this scheme under a one-parameter subgroup action, revealing a decomposition into $n!$ components indexed by permutations.
• To conjecture that the diagonal commutator scheme has exactly two components for any reductive Lie algebra, with implications for Cohen-Macaulayness and reducedness of the commuting variety.
• To provide evidence for the reducedness and Cohen-Macaulayness of the commuting scheme via degeneration to matrix Schubert varieties.

Proposed method

• Define the diagonal commutator scheme $D = \{(X,Y) \in \mathfrak{gl}_n \oplus \mathfrak{gl}_n : [X,Y] \in \mathfrak{h}\}$, where $\mathfrak{h}$ is the Cartan subalgebra of diagonal matrices.
• Use the one-parameter subgroup $\check{\rho}(t) = \operatorname{diag}(1, t, \dots, t^{n-1})$ to define a flat family $D^z$ of schemes parametrized by $z \in \mathbb{C}^\times$, with $D^1 = D$.
• Take the flat limit $D^0 = \lim_{z \to 0} D^z$, which is shown to be a reduced complete intersection defined by $XY$ lower triangular and $YX$ upper triangular.
• Analyze the action of $B_- \times B_+$ (lower and upper triangular matrices) on $D^0$, showing it preserves components and acts transitively on each component $D^0_\pi$.
• Define components $D^0_\pi$ as the closure of the $B_- \times B_+$ orbit of certain pairs involving permutation matrices $\pi$, and prove these are exactly the irreducible components.
• Use matrix Schubert variety techniques and Gröbner basis theory to show that the limit equations are sufficient (i.e., 'Gröbner enough') and no additional equations are needed.

Experimental results

Research questions

• RQ1 Is the diagonal commutator scheme in $\mathfrak{gl}_n$ a reduced complete intersection?
• RQ2 Does the flat degeneration of the diagonal commutator scheme yield a scheme with $n!$ components, each indexed by a permutation?
• RQ3 Can the components of the degenerated scheme be described as closures of $B_- \times B_+$ orbits indexed by permutations?
• RQ4 Does the degeneration provide evidence for the reducedness or Cohen-Macaulayness of the commuting variety?
• RQ5 Are the equations from the trace-perpendicularity of commutators with powers of $X$ and $Y$ sufficient to define the non-commuting component of the diagonal commutator scheme?

Key findings

• The diagonal commutator scheme $D$ is a reduced complete intersection in $\mathfrak{gl}_n \oplus \mathfrak{gl}_n$, with the commuting variety as one component.
• The flat limit $D^0$ of the degeneration is also a reduced complete intersection, defined by $XY$ lower triangular and $YX$ upper triangular.
• The scheme $D^0$ decomposes into exactly $n!$ irreducible components $D^0_\pi$, one for each permutation $\pi \in S_n$, each arising as the closure of a $B_- \times B_+$ orbit.
• The generic fiber component corresponding to the commuting variety specializes to $D^0_\mathbf{1}$ (the identity permutation), possibly with nonreduced structure.
• For $n=3$, the degrees of the components in the degeneration are explicitly computed: $d'_{123} = A^3B^3$, $d'_{213} = 2A^3B + 2A^2B^2 + 2AB^3$, and so on, summing to $(A+B)^6$.
• The authors verify in the $\mathfrak{gl}_3$ case that the equations from the trace-perpendicularity condition (proposition 5) define the non-commuting component of the diagonal commutator scheme, supporting the conjecture that these equations cut out the second component.