96 research outputs found
Opposition diagrams for automorphisms of large spherical buildings
Let be an automorphism of a thick irreducible spherical building
of rank at least with no Fano plane residues. We prove that if
there exist both type and simplices of mapped onto
opposite simplices by , then there exists a type simplex
of mapped onto an opposite simplex by . This property is
called "cappedness". We give applications of cappedness to opposition diagrams,
domesticity, and the calculation of displacement in spherical buildings. In a
companion piece to this paper we study the thick irreducible spherical
buildings containing Fano plane residues. In these buildings automorphisms are
not necessarily capped
Opposition diagrams for automorphisms of small spherical buildings
An automorphism of a spherical building is called
\textit{capped} if it satisfies the following property: if there exist both
type and simplices of mapped onto opposite simplices by
then there exists a type simplex of mapped onto
an opposite simplex by . In previous work we showed that if is
a thick irreducible spherical building of rank at least with no Fano plane
residues then every automorphism of is capped. In the present work we
consider the spherical buildings with Fano plane residues (the \textit{small
buildings}). We show that uncapped automorphisms exist in these buildings and
develop an enhanced notion of "opposition diagrams" to capture the structure of
these automorphisms. Moreover we provide applications to the theory of
"domesticity" in spherical buildings, including the complete classification of
domestic automorphisms of small buildings of types and
A classification of finite homogeneous semilinear spaces
Abstract. A semilinear space S is homogeneous if, whenever the semilinear structures induced on two finite subsets S1 and S2 of S are isomorphic, there is at least one automorphism of S mapping S1 onto S2. We give a complete classification of all finite homogeneous semilinear spaces. Our theorem extends a result of Ronse on graphs and a result of Devillers and Doyen on linear spaces. Key words. Semilinear space, polar space, copolar space, partial geometry, automorphism group, homogeneity. 2000 Mathematics Subject Classification. 05B25, 51E14, 20B25
On Ferri's characterization of the finite quadric Veronesean V24
AbstractWe generalize and complete Ferri's characterization of the finite quadric Veronesean V24 by showing that Ferri's assumptions also characterize the quadric Veroneseans in spaces of even characteristic
Lax embeddings of the Hermitian Unital
In this paper, we prove that every lax generalized Veronesean embedding of the Hermitian unital U of PG(2, L), L a quadratic extension of the field K and |K| ≥ 3, in a PG(d, F), with F any field and d ≥ 7, such that disjoint blocks span disjoint subspaces, is the standard Veronesean embedding in a subgeometry PG(7, K ) of PG(7, F) (and d = 7) or it consists of the projection from a point p ∈ U of U \ {p} from a subgeometry PG(7, K ) of PG(7, F) into a hyperplane PG(6, K ). In order to do so, when |K| > 3 we strongly use the linear representation of the affine part of U (the line at infinity being secant) as the affine part of the generalized quadrangle Q(4, K) (the solid at infinity being non-singular); when |K| = 3, we use the connection of U with the generalized hexagon of order 2
Projective Ring Line Encompassing Two-Qubits
The projective line over the (non-commutative) ring of two-by-two matrices
with coefficients in GF(2) is found to fully accommodate the algebra of 15
operators - generalized Pauli matrices - characterizing two-qubit systems. The
relevant sub-configuration consists of 15 points each of which is either
simultaneously distant or simultaneously neighbor to (any) two given distant
points of the line. The operators can be identified with the points in such a
one-to-one manner that their commutation relations are exactly reproduced by
the underlying geometry of the points, with the ring geometrical notions of
neighbor/distant answering, respectively, to the operational ones of
commuting/non-commuting. This remarkable configuration can be viewed in two
principally different ways accounting, respectively, for the basic 9+6 and 10+5
factorizations of the algebra of the observables. First, as a disjoint union of
the projective line over GF(2) x GF(2) (the "Mermin" part) and two lines over
GF(4) passing through the two selected points, the latter omitted. Second, as
the generalized quadrangle of order two, with its ovoids and/or spreads
standing for (maximum) sets of five mutually non-commuting operators and/or
groups of five maximally commuting subsets of three operators each. These
findings open up rather unexpected vistas for an algebraic geometrical
modelling of finite-dimensional quantum systems and give their numerous
applications a wholly new perspective.Comment: 8 pages, three tables; Version 2 - a few typos and one discrepancy
corrected; Version 3: substantial extension of the paper - two-qubits are
generalized quadrangles of order two; Version 4: self-dual picture completed;
Version 5: intriguing triality found -- three kinds of geometric hyperplanes
within GQ and three distinguished subsets of Pauli operator
Hearing shapes of drums - mathematical and physical aspects of isospectrality
In a celebrated paper '"Can one hear the shape of a drum?"' M. Kac [Amer.
Math. Monthly 73, 1 (1966)] asked his famous question about the existence of
nonisometric billiards having the same spectrum of the Laplacian. This question
was eventually answered positively in 1992 by the construction of noncongruent
planar isospectral pairs. This review highlights mathematical and physical
aspects of isospectrality.Comment: 42 pages, 60 figure
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