{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "6372b98c",
   "metadata": {},
   "source": [
    "# Counting stable quiver representations\n",
    "Given a quiver $Q$ and a stability parameter $z$, let $S_z(d)$ count (absolutely) $z$-stable representations of dimension $d\\in\\mathbb Z^{Q_0}$. It is a polynomial in $q=y^2$.\n",
    "Let $R_z(Q,d)\\subset R(Q,d)$ be the subspace of $z$-semistable representations.\n",
    "Consider the series\n",
    "$$S_{z,c}=\\sum_{\\mu_z(d)=c}S_z(d)x^d,\\qquad A_{z,c}=\\sum_{\\mu_z(d)=c}(-y)^{\\chi(d,d)}\\frac{[R_z(Q,d)]}{[{GL}_d]}x^d\n",
    ",\\qquad c\\in\\mathbb R.$$\n",
    "\n",
    "It was proved in [arXiv:0708.1259](https://arxiv.org/abs/0708.1259) that they satisfy\n",
    "$$A_{z,c}\\circ T\\operatorname{Exp}\\left(\\frac{S_{z,c}}{1-y^2}\\right)=1,\\qquad T(x^d)=(-y)^{\\chi(d,d)}x^d,$$\n",
    "in the quantum affine plane with multiplication \n",
    "$$\\def\\ang#1{\\left<#1\\right>}\n",
    "x^d\\circ x^e=(-y)^{\\ang{d,e}},\\qquad \\ang{d,e}=\\chi(d,e)-\\chi(e,d).$$\n",
    "\n",
    "We compute the series $A_{z,c}$ using the algorithm from [arXiv:math/0204059](https://arxiv.org/abs/math/0204059) (Corollary 5.5) and then apply the above formula to determine the series $S_{z,c}$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5da85255",
   "metadata": {},
   "source": [
    "$\\def\\opr{\\DeclareMathOperator}\n",
    "\\def\\bb{\\mathbb}\n",
    "\\def\\bbZ{\\bb Z}\n",
    "\\def\\bbQ{\\bb Q}\n",
    "\\def\\bbR{\\bb R}\n",
    "\\opr\\Exp{Exp}\n",
    "\\opr\\GL{GL}\n",
    "\\def\\Om{\\Omega}\n",
    "\\def\\hi{\\chi}\n",
    "\\def\\ang#1{\\left<#1\\right>}\n",
    "\\def\\rbr#1{\\left(#1\\right)}\n",
    "$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "2acf3e28",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "import os, sys\n",
    "module_path = os.path.abspath(os.path.join('..'))\n",
    "if module_path not in sys.path:\n",
    "    sys.path.append(module_path)\n",
    "from msinvar import *"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "75f60ac6",
   "metadata": {},
   "source": [
    "## Quiver $1\\to 2$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "0be3f212",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Kronecker quiver\n",
      "Stability function [[1, 0], [1, 1]]\n"
     ]
    }
   ],
   "source": [
    "Q=KroneckerQuiver(1); print(Q)\n",
    "Q.prec([3,3])\n",
    "z=Stability([1,0]); print(z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "c203f4a2",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{(1, 0): 1}"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Q.stable(z, slope=1).dict()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "714a9185",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{(0, 1): 1}"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Q.stable(z, slope=0).dict()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "db303814",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{(1, 1): 1}"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Q.stable(z, slope=1/2).dict()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "65f1a65d",
   "metadata": {
    "tags": []
   },
   "source": [
    "## Kronecker quiver (with 2 arrows)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "77965818",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Kronecker quiver\n"
     ]
    }
   ],
   "source": [
    "Q=KroneckerQuiver(2); print(Q)\n",
    "Q.prec([3,3])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "2ff386db",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{(1, 1): y^2 + 1}"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Q.stable(z, slope=1/2).dict()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1dd6ecaf",
   "metadata": {},
   "source": [
    "## 2-Cyclic quiver"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "b873ce0b",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Cyclic quiver\n"
     ]
    }
   ],
   "source": [
    "Q=CyclicQuiver(2); print(Q)\n",
    "Q.prec((10,10))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "f61617e7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{(1, 1): y^2}"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Q.stable(z, slope=1/2).dict()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "799d5a0d",
   "metadata": {},
   "source": [
    "## 3-Kronecker quiver"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "d3ee06f3",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Kronecker quiver\n"
     ]
    }
   ],
   "source": [
    "Q=KroneckerQuiver(3); print(Q)\n",
    "Q.prec((12,18))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "d3a47b50",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CPU times: user 4.52 s, sys: 7.69 ms, total: 4.53 s\n",
      "Wall time: 4.53 s\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "{(1, 1): y^4 + y^2 + 1,\n",
       " (2, 2): y^10 - y^4,\n",
       " (3, 3): y^20 + y^18 + y^16 - 2*y^12 - y^10,\n",
       " (4,\n",
       "  4): y^34 + y^32 + 3*y^30 + 3*y^28 + 3*y^26 - 2*y^24 - 4*y^22 - 6*y^20 - 3*y^18 + 2*y^14 + 2*y^12,\n",
       " (5,\n",
       "  5): y^52 + y^50 + 3*y^48 + 5*y^46 + 8*y^44 + 10*y^42 + 9*y^40 + 4*y^38 - 6*y^36 - 16*y^34 - 22*y^32 - 20*y^30 - 10*y^28 + 4*y^26 + 13*y^24 + 14*y^22 + 7*y^20 - 3*y^16 - 2*y^14}"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%time I=Q.stable([1,0],1/2) #stable for slope=1/2\n",
    "I.dict([5,5])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "1ee7ab13",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{(1, 1): 3,\n",
       " (2, 2): 0,\n",
       " (3, 3): 0,\n",
       " (4, 4): 0,\n",
       " (5, 5): 0,\n",
       " (6, 6): 0,\n",
       " (7, 7): 0,\n",
       " (8, 8): 0,\n",
       " (9, 9): 0,\n",
       " (10, 10): 0,\n",
       " (11, 11): 0,\n",
       " (12, 12): 0}"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dct={e:c(y=1) for e,c in I.dict().items()}; dct #just the Euler numbers"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "dcd306eb",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CPU times: user 8.64 s, sys: 10.8 ms, total: 8.65 s\n",
      "Wall time: 8.65 s\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "{(2, 3): 13,\n",
       " (4, 6): 49,\n",
       " (6, 9): -28,\n",
       " (8, 12): -5277,\n",
       " (10, 15): -50540,\n",
       " (12, 18): 546995}"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%time I=Q.stable([1,0], slope=2/5)\n",
    "dct={e:c(y=1) for e,c in I.dict().items()}; dct"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a1362b58",
   "metadata": {},
   "source": [
    "## 4-Kronecker quiver"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "352e5750",
   "metadata": {},
   "outputs": [],
   "source": [
    "Q=KroneckerQuiver(4)\n",
    "Q.prec((10,20))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "243bee49",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CPU times: user 14.8 s, sys: 20.4 ms, total: 14.8 s\n",
      "Wall time: 14.8 s\n"
     ]
    }
   ],
   "source": [
    "%time S=Q.stable([1,0], slope=1/3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "6a9d0f90",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{(1, 2): 6,\n",
       " (2, 4): -1,\n",
       " (3, 6): 0,\n",
       " (4, 8): 0,\n",
       " (5, 10): 0,\n",
       " (6, 12): 0,\n",
       " (7, 14): 0,\n",
       " (8, 16): 0,\n",
       " (9, 18): 0,\n",
       " (10, 20): 0}"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dct={e:c(y=1) for e,c in S.dict().items()}; dct"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "45566425",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CPU times: user 14.5 s, sys: 19.2 ms, total: 14.5 s\n",
      "Wall time: 14.5 s\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "{(1, 2): 6,\n",
       " (2, 4): -56,\n",
       " (3, 6): 990,\n",
       " (4, 8): -24000,\n",
       " (5, 10): 682170,\n",
       " (6, 12): -21432264,\n",
       " (7, 14): 721910826,\n",
       " (8, 16): -25590602944,\n",
       " (9, 18): 943269520050,\n",
       " (10, 20): -35857514085080}"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Om=Q.Om(z).restrict(z,1/3)\n",
    "%time Om.subs(y=1).dict()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "18916368",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-3"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d=[1,2]\n",
    "Q.eform(d,d)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "943625bc",
   "metadata": {},
   "source": [
    "## 5-Kronecker quiver"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "57b02709",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Kronecker quiver\n",
      "Stability function [[1, 0], [1, 1]]\n"
     ]
    }
   ],
   "source": [
    "Q=KroneckerQuiver(5); print(Q)\n",
    "Q.prec((10,10))\n",
    "z=Stability([1,0]); print(z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "4d7d2416",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CPU times: user 3.18 s, sys: 6 ms, total: 3.19 s\n",
      "Wall time: 3.19 s\n"
     ]
    }
   ],
   "source": [
    "%time I=Q.stable(z, slope=1/2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "3be0cf32",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{(1, 1): 5,\n",
       " (2, 2): 0,\n",
       " (3, 3): 0,\n",
       " (4, 4): 0,\n",
       " (5, 5): 0,\n",
       " (6, 6): 0,\n",
       " (7, 7): 0,\n",
       " (8, 8): 0,\n",
       " (9, 9): 0,\n",
       " (10, 10): 0}"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dct={e:c(y=1) for e,c in I.dict(Q).items()}; dct"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "b16b03af",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CPU times: user 236 ms, sys: 2.23 ms, total: 238 ms\n",
      "Wall time: 237 ms\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "{(1, 2): 10, (2, 4): -5, (3, 6): 5, (4, 8): -10, (5, 10): 25}"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%time I=Q.stable(z, slope=1/3)\n",
    "dct={e:c(y=1) for e,c in I.dict(W).items()}; dct"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "2174539f",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CPU times: user 236 ms, sys: 2.19 ms, total: 238 ms\n",
      "Wall time: 236 ms\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "{(2, 1): 10, (4, 2): -5, (6, 3): 5, (8, 4): -10, (10, 5): 25}"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%time I=Q.stable(z, slope=2/3)\n",
    "dct={e:c(y=1) for e,c in I.dict().items()}; dct"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "527257fe",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CPU times: user 73.9 ms, sys: 1.17 ms, total: 75.1 ms\n",
      "Wall time: 74.2 ms\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "{(1, 3): 10, (2, 6): -5, (3, 9): 5}"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%time I=Q.stable(z, slope=1/4)\n",
    "dct={e:c(y=1) for e,c in I.dict().items()}; dct"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "5372fee8",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CPU times: user 75.1 ms, sys: 1.47 ms, total: 76.6 ms\n",
      "Wall time: 75.2 ms\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "{(3, 1): 10, (6, 2): -5, (9, 3): 5}"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%time I=Q.stable(z, slope=3/4)\n",
    "dct={e:c(y=1) for e,c in I.dict().items()}; dct"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "99a46151",
   "metadata": {},
   "source": [
    "## 6-Kronecker quiver"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "id": "b315ee86",
   "metadata": {},
   "outputs": [],
   "source": [
    "Q=KroneckerQuiver(6)\n",
    "Q.prec((6,12))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "id": "aa5dbdc0",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CPU times: user 798 ms, sys: 2.79 ms, total: 800 ms\n",
      "Wall time: 799 ms\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "{(1, 2): 15,\n",
       " (2, 4): -15,\n",
       " (3, 6): 36,\n",
       " (4, 8): -141,\n",
       " (5, 10): 660,\n",
       " (6, 12): -3357}"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%time S=Q.stable([1,0], slope=1/3)\n",
    "dct={e:c(y=1) for e,c in S.dict().items()}; dct"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "83b8319e",
   "metadata": {},
   "source": [
    "# Relation to DT invariants"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "38fee796",
   "metadata": {},
   "source": [
    "DT invariants $\\Omega_d(y)$ satisfy the formula\n",
    "$$A_{z,c}=Exp\\left(\\frac{y\\Omega_{z,c}}{1-y^2}\\right),\\qquad \\Omega_{z,c}=\\sum_{\\mu_z(d)=c}\\Omega_d(y) x^d.$$\n",
    "Using the formula for stable objects and assuming that stability $z$ is generic (if $\\mu_z(d)=\\mu_z(e)$, then $d,e$ are proportional), we obtain\n",
    "$$T Exp\\left(\\frac{S_{z,c}}{1-y^2}\\right)=Exp\\left(\\frac{-y\\Omega_{z,c}}{1-y^2}\\right)$$\n",
    "This implies, in particular, that we can express numerical DT invariants $\\Omega_z(d,1)$ in terms of Euler numbers $\\chi_z(d)=S_z(d,1)$ of the moduli spaces of stable representations.\n",
    "\n",
    "Let $d_0$ be the indivisible vector with $\\mu_z(d_0)=c$.\n",
    "Let us define $S_k=S_z(kd_0)$, $\\Omega_k=\\Omega_z(kd_0)$, $N=-\\chi(d_0,d_0)$.\n",
    "Then the above formula can be written in the form\n",
    "$$T\\ Exp\\left(\\frac{\\sum_{k\\ge1}S_kx^k}{1-y^2}\\right)\n",
    "=Exp\\left(\\frac{-y\\sum_{k\\ge1}\\Omega_k x^k}{1-y^2}\\right),\n",
    "\\qquad T(x^k)=(-y)^{-Nk^2}x^k.\n",
    "$$\n",
    "We can express the values $b_k=\\Omega_k(1)$ in terms of values $a_k=S_k(1)$ as\n",
    "$$b_k=f_k(a_1,\\dots,a_k),$$\n",
    "where $f_k$ are polynomials which depend just on $N$.\n",
    "We will compute these polynomials for small $N$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "29d26021",
   "metadata": {},
   "source": [
    "## Calculation of polynomials"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "79537a8b",
   "metadata": {},
   "outputs": [],
   "source": [
    "from msinvar.lambda_rings import adams\n",
    "class TMPoly1(TMPoly):\n",
    "    def adams(self, a, n):\n",
    "        d = {e: adams(c, n) for e, c in a.dict().items()}\n",
    "        return self.Element(self, d)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "2d9eeced",
   "metadata": {},
   "outputs": [],
   "source": [
    "B=RF('y')\n",
    "N=1; m=5\n",
    "R1=TMPoly1(B,m+1,'a')\n",
    "R2=TMPoly(R1,1,'x',prec=m)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "9ebd8c84",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Defining y\n",
      "Defining a0, a1, a2, a3, a4, a5\n",
      "Defining x\n"
     ]
    }
   ],
   "source": [
    "B.inject_variables()\n",
    "R1.inject_variables()\n",
    "R2.inject_variables()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "id": "a05fed24",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "a1*x + a2*x^2 + a3*x^3 + a4*x^4 + a5*x^5"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g=sum(R1.gen(i)*x^i for i in range(1,m+1))\n",
    "g1=(g/(1-y^2)).Exp()\n",
    "g"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "id": "9d1b9237",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{1: a1,\n",
       " 2: (-1/2)*a1 - a2 + (-1/2)*a1^2,\n",
       " 3: a3 + 1/2*a1^2 + 2*a1*a2 + 1/2*a1^3,\n",
       " 4: -a4 + (-1/3)*a1^2 - a1*a2 + (-2)*a2^2 + (-3)*a1*a3 - a1^3 + (-4)*a1^2*a2 + (-2/3)*a1^4,\n",
       " 5: a5 + 5/12*a1^2 + 5/3*a1*a2 + 3/2*a1*a3 + 6*a2*a3 + 4*a1*a4 + 35/24*a1^3 + 5*a1^2*a2 + 10*a1*a2^2 + 15/2*a1^2*a3 + 25/12*a1^4 + 25/3*a1^3*a2 + 25/24*a1^5}"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N=1\n",
    "g2=g1.term_twist(lambda d: (-y)^(-N*d[0]^2)).Log()*(1-y^2)/(-y)\n",
    "dct={e[0]:c.subs_base(y=1) for e,c in g2.dict().items()}; dct"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "id": "95a070c4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{1: a1,\n",
       " 2: (-1/2)*a1 - a2 + 1/2*a1^2,\n",
       " 3: a3 + (-1/2)*a1^2 + (-2)*a1*a2 + 1/2*a1^3,\n",
       " 4: -a4 + 1/3*a1^2 + a1*a2 + 2*a2^2 + 3*a1*a3 - a1^3 + (-4)*a1^2*a2 + 2/3*a1^4,\n",
       " 5: a5 + (-5/12)*a1^2 + (-5/3)*a1*a2 + (-3/2)*a1*a3 + (-6)*a2*a3 + (-4)*a1*a4 + 35/24*a1^3 + 5*a1^2*a2 + 10*a1*a2^2 + 15/2*a1^2*a3 + (-25/12)*a1^4 + (-25/3)*a1^3*a2 + 25/24*a1^5}"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N=-1\n",
    "g2=g1.term_twist(lambda d: (-y)^(-N*d[0]^2)).Log()*(1-y^2)/(-y)\n",
    "dct={e[0]:c.subs_base(y=1) for e,c in g2.dict().items()}; dct"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "id": "61957f8f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{1: -a1,\n",
       " 2: -a2 - a1^2,\n",
       " 3: -a3 - a1^2 + (-4)*a1*a2 + (-2)*a1^3,\n",
       " 4: -a4 + (-2/3)*a1^2 + (-2)*a1*a2 + (-4)*a2^2 + (-6)*a1*a3 + (-4)*a1^3 + (-16)*a1^2*a2 + (-16/3)*a1^4,\n",
       " 5: -a5 + (-5/6)*a1^2 + (-10/3)*a1*a2 + (-3)*a1*a3 + (-12)*a2*a3 + (-8)*a1*a4 + (-35/6)*a1^3 + (-20)*a1^2*a2 + (-40)*a1*a2^2 + (-30)*a1^2*a3 + (-50/3)*a1^4 + (-200/3)*a1^3*a2 + (-50/3)*a1^5}"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N=2\n",
    "g2=g1.term_twist(lambda d: (-y)^(-N*d[0]^2)).Log()*(1-y^2)/(-y)\n",
    "dct={e[0]:c.subs_base(y=1) for e,c in g2.dict().items()}; dct"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "id": "ab37b950",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{1: a1,\n",
       " 2: (-1/2)*a1 - a2 + (-3/2)*a1^2,\n",
       " 3: a3 + 3/2*a1^2 + 6*a1*a2 + 9/2*a1^3,\n",
       " 4: -a4 - a1^2 + (-3)*a1*a2 + (-6)*a2^2 + (-9)*a1*a3 + (-9)*a1^3 + (-36)*a1^2*a2 + (-18)*a1^4,\n",
       " 5: a5 + 5/4*a1^2 + 5*a1*a2 + 9/2*a1*a3 + 18*a2*a3 + 12*a1*a4 + 105/8*a1^3 + 45*a1^2*a2 + 90*a1*a2^2 + 135/2*a1^2*a3 + 225/4*a1^4 + 225*a1^3*a2 + 675/8*a1^5}"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N=3\n",
    "g2=g1.term_twist(lambda d: (-y)^(-N*d[0]^2)).Log()*(1-y^2)/(-y)\n",
    "dct={e[0]:c.subs_base(y=1) for e,c in g2.dict().items()}; dct"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "id": "4b29ae7c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{1: -a1,\n",
       " 2: -a2 + (-2)*a1^2,\n",
       " 3: -a3 + (-2)*a1^2 + (-8)*a1*a2 + (-8)*a1^3,\n",
       " 4: -a4 + (-4/3)*a1^2 + (-4)*a1*a2 + (-8)*a2^2 + (-12)*a1*a3 + (-16)*a1^3 + (-64)*a1^2*a2 + (-128/3)*a1^4,\n",
       " 5: -a5 + (-5/3)*a1^2 + (-20/3)*a1*a2 + (-6)*a1*a3 + (-24)*a2*a3 + (-16)*a1*a4 + (-70/3)*a1^3 + (-80)*a1^2*a2 + (-160)*a1*a2^2 + (-120)*a1^2*a3 + (-400/3)*a1^4 + (-1600/3)*a1^3*a2 + (-800/3)*a1^5}"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N=4\n",
    "g2=g1.term_twist(lambda d: (-y)^(-N*d[0]^2)).Log()*(1-y^2)/(-y)\n",
    "dct={e[0]:c.subs_base(y=1) for e,c in g2.dict().items()}; dct"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "52ca03a8",
   "metadata": {},
   "source": [
    "## Some checks\n",
    "We verify that for the 4-Kronecker quiver we get correct numerical DT invariants for a special stability and special slope:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "id": "6d310036",
   "metadata": {},
   "outputs": [],
   "source": [
    "Q=KroneckerQuiver(4)\n",
    "Q.prec((6,12))\n",
    "z=Stability([1,0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "id": "3da40151",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{(1, 2): 6, (2, 4): -1}"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "S=Q.stable(z,1/3)\n",
    "S1=S.subs(y=1); S1.dict()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "id": "26b65204",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{(1, 2): 6,\n",
       " (2, 4): -56,\n",
       " (3, 6): 990,\n",
       " (4, 8): -24000,\n",
       " (5, 10): 682170,\n",
       " (6, 12): -21432264}"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Q.Om(z).restrict(z,1/3).subs(y=1).dict()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "id": "792218d5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d=[1,2]\n",
    "-Q.eform(d,d)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "id": "cb5d65a5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{1: a1,\n",
       " 2: (-1/2)*a1 - a2 + (-3/2)*a1^2,\n",
       " 3: a3 + 3/2*a1^2 + 6*a1*a2 + 9/2*a1^3,\n",
       " 4: -a4 - a1^2 + (-3)*a1*a2 + (-6)*a2^2 + (-9)*a1*a3 + (-9)*a1^3 + (-36)*a1^2*a2 + (-18)*a1^4,\n",
       " 5: a5 + 5/4*a1^2 + 5*a1*a2 + 9/2*a1*a3 + 18*a2*a3 + 12*a1*a4 + 105/8*a1^3 + 45*a1^2*a2 + 90*a1*a2^2 + 135/2*a1^2*a3 + 225/4*a1^4 + 225*a1^3*a2 + 675/8*a1^5}"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N=3\n",
    "g2=g1.term_twist(lambda d: (-y)^(-N*d[0]^2)).Log()*(1-y^2)/(-y)\n",
    "dct={e[0]:c.subs_base(y=1) for e,c in g2.dict().items()}; dct"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "id": "87186e50",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1 6\n",
      "2 -56\n",
      "3 990\n",
      "4 -24000\n",
      "5 682170\n"
     ]
    }
   ],
   "source": [
    "for i,g in dct.items():\n",
    "    print(i, g.subs(a1=6,a2=-1,a3=0,a4=0,a5=0))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2f8f4fd2",
   "metadata": {},
   "source": [
    "### Further checks"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "860f807f",
   "metadata": {},
   "outputs": [],
   "source": [
    "Q=KroneckerQuiver(4)\n",
    "Q.prec((4,8))\n",
    "z=Stability([1,0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "id": "d55296aa",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1, 2] 0.333333333333 3\n"
     ]
    }
   ],
   "source": [
    "d=[1,2]\n",
    "mu=z.slope(d)\n",
    "N=-Q.eform(d,d)\n",
    "print(d,mu,N)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "id": "85391d84",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{1: 6, 2: -1, 3: 0, 4: 0}"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "S=Q.stable(z,mu).subs(y=1);\n",
    "s={k:int(S([k,2*k])) for k in range(1,5)}; s"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "id": "694c1a38",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{1: -6, 2: 56, 3: -990, 4: 24000}"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Om=Q.Om(z).restrict(z,mu).subs(y=1)\n",
    "om={k:-int(Om([k,2*k])) for k in range(1,5)}; om #need to change the sign because of conventions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "id": "2f417530",
   "metadata": {},
   "outputs": [],
   "source": [
    "R=TMPoly(prec=[4])\n",
    "x=R.gen()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "id": "c8dbeb19",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1 + 6*x + 127*x^2 + 3662*x^3 + 121843*x^4"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F=1\n",
    "for k in range(1,5):\n",
    "    F=F*(1-(-1)^(N*k)*x^k)^(-k*om[k])\n",
    "F"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "id": "6a8f58a8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1 + 6*x + 127*x^2 + 3662*x^3 + 121843*x^4"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "G=1\n",
    "for k in range(1,5):\n",
    "    G=G*(1-(x*F^N)^k)^(-k*s[k]) #functional equation\n",
    "G"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "id": "f832219c",
   "metadata": {},
   "outputs": [],
   "source": [
    "def s2om(k,N,s):\n",
    "    s1=0\n",
    "    for m in divisors(k):\n",
    "        s2=0\n",
    "        for p in Partitions(m):\n",
    "            p=p+[0]\n",
    "            l=[p[i]-p[i+1] for i in range(len(p)-1)]\n",
    "            s2 += prod((-1)^l[i]*binomial(-N*m*(i+1)*s[i+1],l[i]) for i in range(len(l)))\n",
    "        s1 += moebius(k/m)*(-1)^(N*m)*s2\n",
    "    return s1/(N*k^2)            "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "id": "cc0c6268",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{1: -6, 2: 56, 3: -990, 4: 24000}"
      ]
     },
     "execution_count": 102,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "{k:s2om(k,N,s) for k in range(1,5)}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "2c0dcb8d",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "author": {
   "name": "Sergey Mozgovoy"
  },
  "kernelspec": {
   "display_name": "SageMath 9.2",
   "language": "sage",
   "name": "sagemath"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.8.5"
  },
  "title": "Stable quiver representations"
 },
 "nbformat": 4,
 "nbformat_minor": 5
}