[Paper Review] Root systems and diagram calculus. II. Quadratic forms for the Carter diagrams

Rafael Stekolshchik|arXiv (Cornell University)|Oct 27, 2010

Algebraic structures and combinatorial models8 references3 citations

TL;DR

This paper introduces a partial Cartan matrix and dual partial Weyl group for Carter diagrams containing 4-cycles, defining linkage systems via linearly independent root extensions. It shows that linkage systems for $E_6(a_i)$, $E_7(a_i)$, and $D_n(a_i)$ have 2, 1, and 1 components respectively, with sizes 27, 56, and $2n$, matching dimensions of fundamental representations of $E_6$, $E_7$, and $D_n$, with loctets (8-element subsystems) playing a central structural role.

ABSTRACT

For any Carter diagram $Γ$ containing 4-cycle, we introduce the partial Cartan matrix $B_L$, which is similar to the Cartan matrix associated with a Dynkin diagram. A linkage diagram is obtained from $Γ$ by adding one root together with its bonds such that the resulting subset of roots is linearly independent. The linkage diagrams connected under the action of dual partial Weyl group (associated with $B_L$) constitute the linkage system, which is similar to the weight system arising in the representation theory of the semisimple Lie algebras. For Carter diagrams $E_6(a_i)$ and $E_6$ (resp. $E_7(a_i)$ and $E_7$; resp. $D_n(a_i)$ and $D_n$), the linkage system has, respectively, 2, 1, 1 components, each of which contains, respectively, 27, 56, $2n$ elements. Numbers 27, 56 and $2n$ are well-known dimensions of the smallest fundamental representations of semisimple Lie algebras, respectively, for $E_6$, $E_7$ and $D_n$. The 8-cell "spindle-like" linkage subsystems called loctets play the essential role in describing the linkage systems. It turns that weight systems also can be described by means of loctets.

Motivation & Objective

• To develop a diagram calculus for Carter diagrams with 4-cycles using partial Cartan matrices and dual partial Weyl groups.
• To classify linkage systems arising from root extensions that preserve linear independence.
• To explain the appearance of dimensions 27, 56, and $2n$ in linkage systems as analogues of fundamental representation dimensions.
• To establish the role of loctets—8-cell spindle-like subsystems—in organizing linkage systems and weight systems.
• To extend the framework to higher-rank diagrams $D_l(a_k)$ and $D_l$ for $l > 7$, showing consistent structural patterns.

Proposed method

• Define a partial Cartan matrix $B_L$ associated with a Carter diagram $Γ$ containing a 4-cycle, derived from root subsets.
• Introduce the dual partial Weyl group $W_L^ullet$ acting on linkage diagrams to generate linkage systems.
• Use the inverse quadratic form $\mathscr{B}_L^\vee$ to characterize valid linkage diagrams via inequality $\mathscr{B}_L^\vee(\gamma^\vee) < 2$.
• Construct linkage diagrams by adding a root $\gamma$ to $\Gamma$ such that the extended root system remains linearly independent.
• Employ $\beta$-unicolored linkage diagrams and loctets (type $ij \in \{12,13,23\}$) to enumerate components and subsystems.
• Apply projection theorems to extend results from low-rank diagrams ($D_4, D_5, D_6$) to higher-rank cases ($l > 7$).

Experimental results

Research questions

• RQ1 How can a partial Cartan matrix and dual partial Weyl group be defined for Carter diagrams with 4-cycles?
• RQ2 What is the structure of the linkage system generated by extending Carter diagrams with linearly independent roots?
• RQ3 Why do linkage systems for $E_6(a_i)$, $E_7(a_i)$, and $D_n(a_i)$ have sizes 27, 56, and $2n$, respectively?
• RQ4 How do loctets—8-element subsystems—organize the linkage systems and weight systems?
• RQ5 Can the linkage system structure be projected and generalized to higher-rank diagrams such as $D_l(a_k)$ and $D_l$ for $l > 7$?

Key findings

• The linkage system for $E_6(a_i)$ has 2 components, each containing 27 elements, matching the dimension of the fundamental representation of $E_6$.
• The linkage system for $E_7(a_i)$ has 1 component with 56 elements, matching the dimension of the fundamental representation of $E_7$.
• The linkage system for $D_n(a_i)$ has 1 component with $2n$ elements, matching the dimension of the fundamental representation of $D_n$.
• Loctets—8-cell spindle-like subsystems—play a central role in organizing linkage systems, with each component of $E_6(a_i)$, $E_7(a_i)$, and $D_n(a_i)$ containing multiple loctets.
• For $D_l(a_k)$ and $D_l$ with $l > 7$, the linkage system has 1 component with $2l$ elements and 1 loctet, consistent with the $D_n$ pattern.
• The inverse quadratic form $\mathscr{B}_L^\vee$ provides a criterion $\mathscr{B}_L^\vee(\gamma^\vee) < 2$ to identify valid linkage diagrams, enabling systematic enumeration.