Notes on Explicit Block Diagonalization

2014

Electron. J. Combin.

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We inductively construct an explicit (common) orthogonal eigenbasis for the elements of the Bose-Mesner algebra of the Grassmann scheme. The main step is a constructive, linear algebraic interpretation of the Goldman-Rota recurrence for the number of subspaces of a finite vector space. This interpretation shows that the up operator on subspaces has an explicitly given recursive structure. Using this we inductively construct an explicit orthogonal symmetric Jordan basis with respect to the up operator and write down the singular values, i.e., the ratio of the lengths of the successive vectors in the Jordan chains. The collection of all vectors in this basis of a fixed rank forms a (common) orthogonal eigenbasis for the elements of the Bose-Mesner algebra of the Grassmann scheme. We also pose a bijective proof problem on the spanning trees of the Grassmann graphs.

Section: Introductionmentioning

confidence: 63%

“…More generally, we can explicitly block diagonalize End GL(n,q) (V (B q (n))). We refer to [19] for details.…”

Section: Orthogonal Symmetric Jordan Basismentioning

confidence: 99%

The Goldman-Rota Identity and the Grassmann Scheme

2014

Electron. J. Combin.

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“…For a subspace analog (or q-analog) of Theorem 2.1, see [1,10]. For explicit constructions of orthogonal SJB's of V (B(n)) and V (B q (n)) (the subspace analog of V (B(n))) see [9,11].…”

Section: Theorem 21 (I) There Exists An Orthogonal Sjb J(n) Of V (B(n))mentioning

confidence: 99%

“…Taking dimensions on both sides of (20) yields a linear algebraic interpretation to identity (10) above. Certain problems on the generalized Boolean algebra B X (n) can be reduced to corresponding problems on the Boolean algebra B(n) via the basis K X (n).…”

Section: Consider the Identitymentioning

confidence: 99%

See 1 more Smart Citation

Wreath Product Action on Generalized Boolean Algebras

Mishra

2015

Electron. J. Combin.

Self Cite

A Combinatorial Determinant Dual to the Group Determinant

Mishra

2015

ELA

We define the commuting algebra determinant of a finite group action on a finite set, a notion dual to the group determinant of Dedekind. We give the following combinatorial example of a commuting algebra determinant. Let $\Bq(n)$ denote the set of all subspaces of an $n$-dimensional vector space over $\Fq$. The {\em type} of an ordered pair $(U,V)$ of subspaces, where $U,V\in \Bq(n)$, is the ordered triple $(\mbox{dim }U, \mbox{dim }V, \mbox{dim }U\cap V)$ of nonnegative integers. Assume that there are independent indeterminates corresponding to each type. Let $X_q(n)$ be the $\Bq(n)\times \Bq(n)$ matrix whose entry in row $U$, column $V$ is the indeterminate corresponding to the type of $(U,V)$. We factorize the determinant of $X_q(n)$ into irreducible polynomials.