[Paper Review] Upper triangular matrices and Billiard Arrays

Yang Yang|arXiv (Cornell University)|Aug 18, 2015

Cellular Automata and Applications8 references3 citations

TL;DR

This paper establishes a correspondence between very good upper triangular matrices and isomorphism classes of Billiard Arrays on a (d+1)-dimensional vector space over a field F. It shows that such matrices induce three totally opposite flags, which in turn generate a Billiard Array whose B-values (scalar parameters) are explicitly computed in terms of the matrix entries. The key contribution is a bijective correspondence between equivalence classes of very good upper triangular matrices (under diagonal scaling) and isomorphism classes of Billiard Arrays.

ABSTRACT

Fix a nonnegative integer $d$, a field $\mathbb{F}$, and a vector space $V$ over $\mathbb{F}$ with dimension $d+1$. Let $T$ denote an invertible upper triangular matrix in ${ m Mat}_{d+1}(\mathbb{F})$. Using $T$ we construct three flags on $V$. We find a necessary and sufficient condition on $T$ for these three flags to be totally opposite. In this case, we use these three totally opposite flags to construct a Billiard Array $B$ on $V$. It is known that $B$ is determined up to isomorphism by a certain triangular array of scalar parameters called the $B$-values. We compute these $B$-values in terms of the entries of $T$. We describe the set of isomorphism classes of Billiard Arrays in terms of upper triangular matrices.

Motivation & Objective

• To characterize when an invertible upper triangular matrix induces three totally opposite flags on a (d+1)-dimensional vector space.
• To compute the B-values of the Billiard Array constructed from such flags in terms of the matrix entries.
• To establish a bijection between equivalence classes of very good upper triangular matrices (under diagonal scaling) and isomorphism classes of Billiard Arrays.
• To provide an explicit example where the B-values are all equal to q⁻¹ using q-binomial coefficients.

Proposed method

• Construct three flags on a vector space V of dimension d+1 from an invertible upper triangular matrix T: one from the standard basis, one from its reverse, and one from the image basis under T.
• Define a matrix T as 'very good' if all its principal submatrices are invertible, which ensures the three flags are totally opposite.
• Use the theory of totally opposite flags to construct a Billiard Array B on V, which is uniquely determined up to isomorphism by its B-values.
• Compute the B-values explicitly using a formula involving determinants of submatrices of T, specifically det(T[i,j]) for 0 ≤ i ≤ j ≤ d.
• Define an equivalence relation ∼ on the set of very good upper triangular matrices via left and right multiplication by invertible diagonal matrices.
• Establish a commutative diagram showing that the map from equivalence classes of matrices to isomorphism classes of Billiard Arrays is a bijection.

Experimental results

Research questions

• RQ1 Under what condition on an invertible upper triangular matrix T are the three flags it induces on V totally opposite?
• RQ2 How can the B-values of the Billiard Array constructed from such flags be expressed in terms of the entries of T?
• RQ3 What is the relationship between the equivalence classes of very good upper triangular matrices (under diagonal scaling) and the isomorphism classes of Billiard Arrays?
• RQ4 Is there a canonical matrix representation for each isomorphism class of Billiard Arrays, and if so, how is it constructed?
• RQ5 What happens to the B-values when the matrix entries are q-binomial coefficients?

Key findings

• A necessary and sufficient condition for three flags induced by an invertible upper triangular matrix T to be totally opposite is that T is 'very good', meaning all its principal submatrices are invertible.
• The B-values of the Billiard Array constructed from such flags are given by a formula involving the determinants of submatrices of T, specifically B-values are determined by det(T[i,j]) for 0 ≤ i ≤ j ≤ d.
• For the matrix T with entries T_{ij} = {j rack i}_q (q-binomial coefficients), all B-values of the corresponding Billiard Array are equal to q⁻¹.
• The set of isomorphism classes of Billiard Arrays on V is in bijection with the set of equivalence classes of very good upper triangular matrices under the equivalence relation T ∼ T' iff T' = H T K for invertible diagonal matrices H, K.
• The determinant of the submatrix T[i,j] (rows i to j, columns i to j) is given by det(T[i,j]) = q^{i(j-i)(j-i+1)/2}.
• When q = 1, the matrix entries become binomial coefficients, and the corresponding Billiard Array has all B-values equal to 1.