{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Nonexistence of a $Q$-polynomial association scheme with Krein array $\\{24, 20, 36/11; 1, 30/11, 24\\}$\n",
"\n",
"We will show that a $Q$-polynomial association scheme with Krein array $\\{24, 20, 36/11; 1, 30/11, 24\\}$ does not exist."
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"%display latex\n",
"import drg"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Such a scheme would have $225$ vertices."
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/plain": [
"225"
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"p = drg.QPolyParameters([24, 20, 36/11], [1, 30/11, 24])\n",
"p.order()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The scheme has two $Q$-polynomial orderings $(0, 1, 2, 3)$ and $(0, 3, 2, 1)$, so we have $q^3_{11} = q^1_{13} = q^1_{31} = q^1_{33} = q^3_{13} = q^3_{31} = 0$."
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/plain": [
"0: [ 1 0 0 0]\n",
" [ 0 24 0 0]\n",
" [ 0 0 176 0]\n",
" [ 0 0 0 24]\n",
"\n",
"1: [ 0 1 0 0]\n",
" [ 1 3 20 0]\n",
" [ 0 20 132 24]\n",
" [ 0 0 24 0]\n",
"\n",
"2: [ 0 0 1 0]\n",
" [ 0 30/11 18 36/11]\n",
" [ 1 18 139 18]\n",
" [ 0 36/11 18 30/11]\n",
"\n",
"3: [ 0 0 0 1]\n",
" [ 0 0 24 0]\n",
" [ 0 24 132 20]\n",
" [ 1 0 20 3]"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"p.kreinParameters()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The intersection numbers are integral and nonnegative."
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/plain": [
"0: [ 1 0 0 0]\n",
" [ 0 88 0 0]\n",
" [ 0 0 88 0]\n",
" [ 0 0 0 48]\n",
"\n",
"1: [ 0 1 0 0]\n",
" [ 1 43 32 12]\n",
" [ 0 32 32 24]\n",
" [ 0 12 24 12]\n",
"\n",
"2: [ 0 0 1 0]\n",
" [ 0 32 32 24]\n",
" [ 1 32 43 12]\n",
" [ 0 24 12 12]\n",
"\n",
"3: [ 0 0 0 1]\n",
" [ 0 22 44 22]\n",
" [ 0 44 22 22]\n",
" [ 1 22 22 3]"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"p.pTable()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The parameters do not exceed the absolute bound."
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/plain": [
"{}"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"p.check_absoluteBound()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let $w, x, y, z$ be vertices such that $z$ is in relation $1$ with $x$ and $y$, and $w, x, y$ are mutually in relation $3$. Note that we have $p^3_{11} = 22$ and $p^3_{33} = 3$, so such vertices must exist. We first compute the triple intersection numbers with respect to $x, y, z$. The parameters $\\alpha$ and $\\beta$ will denote the numbers of vertices in relations $(2, 1, 1)$ and $(2, 2, 1)$, respectively, to $x, y, z$."
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/plain": [
"0: [0 0 0 0]\n",
" [0 0 0 0]\n",
" [0 0 0 0]\n",
" [0 1 0 0]\n",
"\n",
"1: [ 0 0 0 0]\n",
" [ 1 -7/2*alpha - beta + 97 alpha - 16 5/2*alpha + beta - 60]\n",
" [ 0 alpha beta + 8 -alpha - beta + 36]\n",
" [ 0 5/2*alpha + beta - 54 -alpha - beta + 40 -3/2*alpha + 36]\n",
"\n",
"2: [ 0 0 0 0]\n",
" [ 0 alpha beta + 8 -alpha - beta + 36]\n",
" [ 0 beta -alpha - 7/2*beta + 52 alpha + 5/2*beta - 30]\n",
" [ 0 -alpha - beta + 32 alpha + 5/2*beta - 28 -3/2*beta + 18]\n",
"\n",
"3: [ 0 1 0 0]\n",
" [ 0 5/2*alpha + beta - 54 -alpha - beta + 40 -3/2*alpha + 36]\n",
" [ 0 -alpha - beta + 32 alpha + 5/2*beta - 28 -3/2*beta + 18]\n",
" [ 0 -3/2*alpha + 33 -3/2*beta + 12 3/2*alpha + 3/2*beta - 42]"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"p.tripleEquations(3, 1, 1, params={\"alpha\": (2, 1, 1), \"beta\": (2, 2, 1)})"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"From $[1\\ 1\\ 3] \\ge 0$ and $[3\\ 3\\ 2] \\ge 0$, it follows that $60 - 5\\alpha/2 \\le \\beta \\le 8$. Using this and $[3\\ 3\\ 1] \\ge 0$, we obtain $20.8 \\le a \\le 22$, but since $\\alpha$ and $\\beta$ must clearly be even integers, it follows that there are two solutions with $\\alpha = 22$ and $\\beta \\in \\{6, 8\\}$. In both cases we have $[3\\ 3\\ 1] = 0$, implying that $w$ cannot be in relation $1$ with $z$ for any choice of $w, x, y, z$ as above.\n",
"\n",
"We now compute the triple intersection numbers with respect to $w, x, y$. The parameters $\\gamma$ and $\\delta$ will denote the numbers of vertices in relations $(1, 1, 2)$ and $(1, 2, 2)$, respectively, to $w, x, y$."
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/plain": [
"0: [0 0 0 0]\n",
" [0 0 0 0]\n",
" [0 0 0 0]\n",
" [0 0 0 1]\n",
"\n",
"1: [ 0 0 0 0]\n",
" [ 0 -delta - 7/2*gamma + 66 gamma delta + 5/2*gamma - 44]\n",
" [ 0 gamma delta -delta - gamma + 44]\n",
" [ 0 delta + 5/2*gamma - 44 -delta - gamma + 44 -3/2*gamma + 22]\n",
"\n",
"2: [ 0 0 0 0]\n",
" [ 0 gamma delta -delta - gamma + 44]\n",
" [ 0 delta -7/2*delta - gamma + 66 5/2*delta + gamma - 44]\n",
" [ 0 -delta - gamma + 44 5/2*delta + gamma - 44 -3/2*delta + 22]\n",
"\n",
"3: [ 0 0 0 1]\n",
" [ 0 delta + 5/2*gamma - 44 -delta - gamma + 44 -3/2*gamma + 22]\n",
" [ 0 -delta - gamma + 44 5/2*delta + gamma - 44 -3/2*delta + 22]\n",
" [ 1 -3/2*gamma + 22 -3/2*delta + 22 3/2*delta + 3/2*gamma - 42]"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"p.tripleEquations(3, 3, 3, params={\"gamma\": (1, 1, 2), \"delta\": (1, 2, 2)})"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"From $[1\\ 3\\ 3] \\ge 0$ and $[2\\ 3\\ 3] \\ge 0$, it follows that $\\gamma, \\delta \\le 44/3$, while $[3\\ 3\\ 3] \\ge 0$ implies $\\gamma + \\delta \\ge 28$. Since $\\gamma$ and $\\delta$ must clearly be even integers, it follows that there is only one solution with $\\gamma = \\delta = 14$ and therefore $[1\\ 1\\ 1] = 3$. This would imply the existence of a vertex $z$ as above that is in relation $1$ with $w$ - a contradiction! We thus conclude that a $Q$-polynomial association scheme with Krein array $\\{24, 20, 36/11; 1, 30/11, 24\\}$ **does not exist**."
]
}
],
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