Short history of signed exceptional sequences

Abstract

Whereas exceptional sequences have a long history with many well-known connections to combinatorics, signed exceptional sequences are relatively recent. The authors introduced this concept in 2017 [19], although it was retroactively realized that the category of noncrossing partitions [24] is a special case of this construction. Buan and Marsh [4] have introduced the concept of $τ$-exceptional sequences to generalize the definitions and theorems to all finite dimensional algebras. This short paper is the story of the original concept of signed exceptional sequences for hereditary algebras and how it developed out of the two authors' study of algebraic K-theory, link invariants, and cluster combinatorics.

Figures (14)

• Figure 1: These are two "second order Steinberg relations". On the left we have $u,v,w\in\pi$, $i,j,k$ distinct, $\ell\neq j,k,m$ and $m\neq i,j$. We also require $x_{ik}^{uv}\neq x_{\ell m}^w$ since $[x,x]$ is not a reduced word. The figure on the right has similar constraints plus the restriction that $x_{ij}^u,x_{k\ell}^v,x_{mn}^w$ are distinct.
• Figure 2: This is one more second order relation which will be used later. The colors will be explained.
• Figure 3: The domain $D(\varphi)$ is shown in blue. These are the points with coordinates $(x,x,y)$. $D(\psi)$ is in red. These are the points $(x,y,x)$ where $x\le y$. The figure shows the normalized dimension vectors: normalized by dividing by the sum $\sum v_i$.
• Figure 4: This is the $g$-vector fan FZ4PPPP for $A_3$ in ${\mathbb{R}}^3$ intersected with the unit sphere $S^2$ and stereographically projected to the plane. Thus, hyperplanes, such as $D(S_1)$ become circles in this figure. The half-plane $D(P_2)$ becomes a semi-circle. This figure is homeomorphic to the figure in Example \ref{['eg: A3 cluster tilting objects']}. We call this the semi-invariant picture for $A_3$. This is also the representation theoretic version of the "second order Steinberg relation" shown in Figure \ref{['Fig: A3 picture with xij']}.
• Figure 5: The picture group of this picture is the Heisenberg group $H=\left< x,y,z\,|\, [x,z],[y,z],[x,y]z^{-1}\right>$. This is a torsion-free nilpotent group since $[x,y]=z$ is central making $H$ a central extension of ${\mathbb{Z}}$ by ${\mathbb{Z}}^2$. This is isomorphic to the unipotent upper triangular matrix group $U(A_2)$ discussed below.

Theorems & Definitions (40)

• Definition 1.1
• Example 1.2
• Example 1.3
• Theorem 1.4
• Theorem 1.5
• Definition 1.6
• Theorem 1.7
• Example 2.1
• Example 2.2
• Theorem 2.3