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Hau-wen Huang

Publications and source records attributed to Hau-wen Huang.

9 recordsLinked to original sources
arXiv2015

Finite-dimensional irreducible modules of the universal Askey-Wilson algebra

Since the introduction of Askey-Wilson algebras by Zhedanov in 1991, the classification of the finite-dimensional irreducible modules of Askey-Wilson algebras remains open. A universal analog $\triangle_q$ of the Askey-Wilson algebras was recently studied. In this paper, we consider a family of infinite-dimensional $\triangle_q$-modules. By the universal property of these $\triangle_q$-modules, we classify the finite-dimensional irreducible $\triangle_q$-modules when $q$ is not a root of unity.

math.RA↗
arXiv2012

Lit-only sigma-game on nondegenerate graphs

A configuration of the lit-only $σ$-game on a graph $Γ$ is an assignment of one of two states, {\it on} or {\it off}, to each vertex of $Γ.$ Given a configuration, a move of the lit-only $σ$-game on $Γ$ allows the player to choose an {\it on} vertex $s$ of $Γ$ and change the states of all neighbors of $s.$ Given an integer $k$, the underlying graph $Γ$ is said to be $k$-lit if for any configuration, the number of {\it on} vertices can be reduced to at most $k$ by a finite sequence of moves. We give a description of the orbits of the lit-only $σ$-game on nondegenerate graphs $Γ$ which are not line graphs. We show that these graphs $Γ$ are 2-lit and provide a linear algebraic criterion for $Γ$ to be 1-lit.

math.CO↗
arXiv2012

Two-lit trees for lit-only sigma-game

A configuration of the lit-only $σ$-game on a finite graph $Γ$ is an assignment of one of two states, on or off, to all vertices of $Γ.$ Given a configuration, a move of the lit-only $σ$-game on $Γ$ allows the player to choose an on vertex $s$ of $Γ$ and change the states of all neighbors of $s.$ Given any integer $k$, we say that $Γ$ is $k$-lit if, for any configuration, the number of on vertices can be reduced to at most $k$ by a finite sequence of moves. Assume that $Γ$ is a tree with a perfect matching. We show that $Γ$ is 1-lit and any tree obtained from $Γ$ by adding a new vertex on an edge of $Γ$ is 2-lit.

math.CO↗
arXiv2012

Correction of a theorem on the symmetric group generated by transvections

Let $V$ denote a vector space over two-element field $\mathbb F_2$ with finite positive dimension and endowed with a symplectic form $B.$ Let ${\rm SL}(V)$ denote the special linear group of $V.$ Let $S$ denote a subset of $V.$ Define $Tv(S)$ as the subgroup of ${\rm SL}(V)$ generated by the transvections with direction $α$ for all $α\in S.$ Define $G(S)$ as the graph whose vertex set is $S$ and where $α,β\in S$ are connected whenever $B(α,β)=1.$ A well-known theorem states that under the assumption that $S$ spans $V,$ the following (i), (ii) are equivalent: (i) $Tv(S)$ is isomorphic to a symmetric group. (ii) $G(S)$ is a claw-free block graph. We give an example which shows that this theorem is not true. We give a modification of this theorem as follows. Assume that $S$ is a linearly independent set of $V$ and no element of $S$ is in the radical of $V.$ Then the above (i), (ii) are equivalent.

math.GR↗
arXiv2011

The classification of Leonard triples of QRacah type

Let $\mathbb{K}$ denote an algebraically closed field. Let $V$ denote a vector space over $\mathbb{K}$ with finite positive dimension. By a Leonard triple on $V$ we mean an ordered triple of linear transformations in ${\rm End}(V)$ such that for each of these transformations there exists a basis of $V$ with respect to which the matrix representing that transformation is diagonal and the matrices representing the other two transformations are irreducible tridiagonal. There is a family of Leonard triples said to have QRacah type. This is the most general type of Leonard triple. We classify the Leonard triples of QRacah type up to isomorphism. We show that any Leonard triple of QRacah type satisfies the $\mathbb{Z}_3$-symmetric Askey-Wilson relations.

math.RA↗
arXiv2010

Combinatorial representations of Coxeter groups over a field of two elements

Let $W$ denote a simply-laced Coxeter group with $n$ generators. We construct an $n$-dimensional representation $ϕ$ of $W$ over the finite field $F_2$ of two elements. The action of $ϕ(W)$ on $F_2^n$ by left multiplication is corresponding to a combinatorial structure extracted and generalized from Vogan diagrams. In each case W of types A, D and E, we determine the orbits of $F_2^n$ under the action of $ϕ(W)$, and find that the kernel of $ϕ$ is the center $Z(W)$ of $W.$

math.RT↗
arXiv2009

The flipping puzzle on a graph

Let $S$ be a connected graph which contains an induced path of $n-1$ vertices, where $n$ is the order of $S.$ We consider a puzzle on $S$. A configuration of the puzzle is simply an $n$-dimensional column vector over $\{0, 1\}$ with coordinates of the vector indexed by the vertex set $S$. For each configuration $u$ with a coordinate $u_s=1$, there exists a move that sends $u$ to the new configuration which flips the entries of the coordinates adjacent to $s$ in $u.$ We completely determine if one configuration can move to another in a sequence of finite steps.

math.CO↗
arXiv2009

The edge-flipping group of a graph

Let $X=(V,E)$ be a finite simple connected graph with $n$ vertices and $m$ edges. A configuration is an assignment of one of two colors, black or white, to each edge of $X.$ A move applied to a configuration is to select a black edge $ε\in E$ and change the colors of all adjacent edges of $ε.$ Given an initial configuration and a final configuration, try to find a sequence of moves that transforms the initial configuration into the final configuration. This is the edge-flipping puzzle on $X,$ and it corresponds to a group action. This group is called the edge-flipping group $\mathbf{W}_E(X)$ of $X.$ This paper shows that if $X$ has at least three vertices, $\mathbf{W}_E(X)$ is isomorphic to a semidirect product of $(\mathbb{Z}/2\mathbb{Z})^k$ and the symmetric group $S_n$ of degree $n,$ where $k=(n-1)(m-n+1)$ if $n$ is odd, $k=(n-2)(m-n+1)$ if $n$ is even, and $\mathbb{Z}$ is the additive group of integers.

math.CO↗