Let $Dgeq 3$ denote an integer. For any $xin mathbb F_2^D$ let $w(x)$ denote the Hamming weight of $x$. Let $X$ denote the subspace of $mathbb F_2^D$ consisting of all $xin mathbb F_2^D$ with even $w(x)$. The $D$-dimensional halved cube $frac{1}{2}H(
D,2)$ is a finite simple connected graph with vertex set $X$ and $x,yin X$ are adjacent if and only if $w(x-y)=2$. Fix a vertex $xin X$. The Terwilliger algebra $mathcal T=mathcal T(x)$ of $frac{1}{2}H(D,2)$ with respect to $x$ is the subalgebra of ${rm Mat}_X(mathbb C)$ generated by the adjacency matrix $A$ and the dual adjacency matrix $A^*=A^*(x)$ where $A^*$ is a diagonal matrix with $$ A^*_{yy}=D-2w(x-y) qquad hbox{for all $yin X$}. $$ In this paper we decompose the standard $mathcal T$-module into a direct sum of irreducible $mathcal T$-modules.
Assume that $mathbb F$ is an algebraically closed field with characteristic zero. The universal Racah algebra $Re$ is a unital associative $mathbb F$-algebra generated by $A,B,C,D$ and the relations state that $[A,B]=[B,C]=[C,A]=2D$ and each of $$ [A
,D]+AC-BA, qquad [B,D]+BA-CB, qquad [C,D]+CB-AC $$ is central in $Re$. The universal additive DAHA (double affine Hecke algebra) $mathfrak H$ of type $(C_1^vee,C_1)$ is a unital associative $mathbb F$-algebra generated by ${t_i}_{i=0}^3$ and the relations state that begin{gather*} t_0+t_1+t_2+t_3 = -1, hbox{$t_i^2$ is central for all $i=0,1,2,3$}. end{gather*} Any $mathfrak H$-module can be considered as a $Re$-module via the $mathbb F$-algebra homomorphism $Reto mathfrak H$ given by begin{eqnarray*} A &mapsto & frac{(t_0+t_1-1)(t_0+t_1+1)}{4}, B &mapsto & frac{(t_0+t_2-1)(t_0+t_2+1)}{4}, C &mapsto & frac{(t_0+t_3-1)(t_0+t_3+1)}{4}. end{eqnarray*} Let $V$ denote a finite-dimensional irreducible $mathfrak H$-module. In this paper we show that $A,B,C$ are diagonalizable on $V$ if and only if $A,B,C$ act as Leonard triples on all composition factors of the $Re$-module $V$.
Let $W$ denote a simply-laced Coxeter group with $n$ generators. We construct an $n$-dimensional representation $phi$ of $W$ over the finite field $F_2$ of two elements. The action of $phi(W)$ on $F_2^n$ by left multiplication is corresponding to a c
ombinatorial 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 $phi(W)$, and find that the kernel of $phi$ is the center $Z(W)$ of $W.$
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 $epsilonin E$ an
d change the colors of all adjacent edges of $epsilon.$ 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 $(mathbb{Z}/2mathbb{Z})^k$ 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.
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.
