Zhenheng Li

THE RENNER MONOIDS AND CELL DECOMPOSITIONS OF THE SYMPLECTIC ALGEBRAIC MONOIDS

In this paper we explicitly determine the Renner monoid ℛ and the cross section lattice Λ of the symplectic algebraic monoid MSpn in terms of the Weyl group and the concept of admissible sets; it turns out that ℛ is a submonoid of ℛn, the Renner monoid of the whole matrix monoid Mn, and that Λ is a sublattice of Λn, the cross section lattice of Mn. Cell decompositions in algebraic geometry are usually obtained by the method of [1]. We give a more direct definition of cells for MSpn in terms of the B × B-orbits, where B is a Borel subgroup of the unit group G of MSpn. Each cell turns out to be the intersection of MSpn with a cell of Mn. We also show how to obtain these cells using a carefully chosen one parameter subgroup.

REPRESENTATIONS OF THE RENNER MONOID

We describe irreducible representations and character formulas of the Renner monoids for reductive monoids, which generalizes the Munn-Solomon representation theory of rook monoids to any Renner monoids. The type map and polytope associated with reductive monoids play a crucial role in our work. It turns out that the irreducible representations of certain parabolic subgroups of the Weyl groups determine the complete set of irreducible representations of the Renner monoids. An analogue of the Munn-Solomon formula for calculating the character of the Renner monoids, in terms of the characters of the parabolic subgroups, is shown.

Representation and Interpretation of Sketches in Mechanical Design: Experimental and Theoretical Approaches

This paper aims to establish a theoretical foundation for representing and interpreting free-hand design sketches throughout the conceptual design process. Both experimental and theoretical approaches are used. In using the experimental approach, one case study from a book and one case study from an automobile assembly system manufacturer are used to illustrate the characteristics of design sketches. These characteristics provide the requirements for models of sketch representation and interpretation. In using the theoretical approach, a mathematical structure of design sketches is established. This mathematical structure can naturally and logically model the evolving sketches generated in the design process, through integrating the strengths of set theory and mereology. Based on the results of these two approaches, a design sketch language is developed to be a formal foundation of sketch representation and interpretation.Copyright

REPRESENTATIONS OF THE SYMPLECTIC ROOK MONOID

In this paper, we concern representations of symplectic rook monoids R. First, an algebraic description of R as a submonoid of a rook monoid is obtained. Second, we determine irreducible representations of R in terms of the irreducible representations of certain symmetric groups and those of the symplectic Weyl group W. We then give the character formula of R using the character of W and that of the symmetric groups. A practical algorithm is provided to make the formula user-friendly. At last we show that the Munn character table of R is a block upper triangular matrix.

The Symplectic Monoid

In this paper, we describe explicitly the symplectic monoid ℳ and its Renner monoid ℛ using elementary methods. We refine the Bruhat–Renner decomposition of ℳ and analyze in detail the length function on ℛ. We then show that every element of ℳ has a unique canonical form decomposition, which is an analogue of the canonical form of elements in Chevalley groups. We also compute the order of ℳ over a finite field, and as a consequence we obtain a new combinatorial identity.

Cell Decompositions of the Special Orthogonal Algebraic Monoids

Abstract Let MSO n (n is even) be the special orthogonal algebraic monoid, B a Borel subgroup of its unit group. We take a more direct definition of cells for MSO n in terms of B × B-orbits and explicitly determine the cell decompositions of MSO n . It turns out that each cell is the intersection of MSO n with a cell of M n (K), the monoid of all n×n matrices over an algebraically closed field K.

Lessons From Experimental Computer Assisted Sketching Tool for Planar Mechanisms

This paper describes the authors’ experiences in implementing a prototype to explore the technical and research challenges in developing a mechanical design system based on the sketching metaphor. This includes basic approaches, methodology, framework, and research issues. Since the objective of the developing the prototype was to identify the issues, a simple, yet representative design domain of conceptual design of planar mechanisms was chosen. The prototype has the functionality to support users creating skeleton sketches of planar mechanisms on a tablet PC. The freehand sketches are then interpreted to derive the structure of the mechanism consisting of linkages and joints. The paper describes the approach for deriving components from skeleton diagrams and outlines the research challenges in way of creating design tools based on the sketching metaphor.Copyright

Augmenting Mechanical CAD With Pen and Tablet

Most designers and engineers tasked with creating concepts for a new machine or mechanical device do not first reach for their CAD station; they reach for pen and paper. However, designs generated on pen and paper often need to be redone in CAD systems and the prevalent belief is that capturing and processing these conceptual designs digitally from the start would save time, reduce errors, cut costs, prevent loss of prior concepts, add traceability, improve innovation and preserve expert knowledge. As preparation for a collaborative project building a tool to assist in the conceptual design of manufacturing/assembly systems, a broad current state of the art survey on pen-, table- and sketch-based design was done. This paper summarises part of that survey focussing on automated and/or interactive understanding of sketches of 3D systems, pen-based CAD, and existing systems exhibiting some of these technologies.Copyright