{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "897a772f-1c3e-4af9-844f-e584af67fe4b",
   "metadata": {},
   "outputs": [],
   "source": [
    "import os \n",
    "import sys\n",
    "sys.path.insert(0, os.path.abspath('../..'))\n",
    "\n",
    "import sympy as sp\n",
    "import numpy as np\n",
    "import itertools as it\n",
    "import pysymmpol as sy\n",
    "import pysymmpol.utils as ut\n",
    "from IPython.display import display, Latex"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3edb64f5-5599-444f-be6d-c7b80fafb61e",
   "metadata": {},
   "source": [
    "# Class: HallLittlewoodPolynomial"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c389d95f-87a7-4fd4-b619-1f3c8cce9de7",
   "metadata": {},
   "source": [
    "We finally consider the Hall-Littlewood polynomials. We have chosen the deformation parameter as $Q$, instead $t$ -- since $t$ is related to our Miwa coordianates. "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0a1fa7db-9a53-4baf-a16b-1feab057eb36",
   "metadata": {},
   "source": [
    "## Basic Usage"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "bba74741-a4bf-4a31-aab9-6307d142d6a8",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Define the deformation parameter\n",
    "Q = sp.Symbol('Q')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "af06fb41-e245-4062-8f59-012e433465eb",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Build Young diagrams for a given partition\n",
    "part = (4,2)\n",
    "yg = sy.YoungDiagram(part)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "84e58fc2-a79d-4703-aa82-0354abe2731f",
   "metadata": {},
   "outputs": [],
   "source": [
    "hl = sy.HallLittlewoodPolynomial(yg)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "2b6e0a73-fdd4-4b8e-8177-04a89286af3f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - Q x_{1}^{3} x_{2}^{3} + x_{1}^{4} x_{2}^{2} + x_{1}^{3} x_{2}^{3} + x_{1}^{2} x_{2}^{4}$"
      ],
      "text/plain": [
       "-Q*x1**3*x2**3 + x1**4*x2**2 + x1**3*x2**3 + x1**2*x2**4"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Let us build the coordinates\n",
    "n = 2\n",
    "x = ut.create_x_coord(n)\n",
    "\n",
    "# Explicit expression for the Hall-Littlewood polynomial\n",
    "hl.explicit(x, Q).expand()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "cae7db2c-12f3-4841-b80c-ffd372868ecf",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\operatorname{Poly}{\\left( x_{1}^{4}x_{2}^{2} - x_{1}^{3}x_{2}^{3}Q + x_{1}^{3}x_{2}^{3} + x_{1}^{2}x_{2}^{4}, x_{1}, x_{2}, Q, domain=\\mathbb{Q} \\right)}$"
      ],
      "text/plain": [
       "Poly(x1**4*x2**2 - x1**3*x2**3*Q + x1**3*x2**3 + x1**2*x2**4, x1, x2, Q, domain='QQ')"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# As before \n",
    "hl.explicit(x, Q, True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "88c6ad90-c9ac-49f5-b7a8-4330e794cc51",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - Q^{3} x_{1}^{2} x_{2}^{2} x_{3}^{2} + Q^{2} x_{1}^{3} x_{2}^{2} x_{3} + Q^{2} x_{1}^{3} x_{2} x_{3}^{2} + Q^{2} x_{1}^{2} x_{2}^{3} x_{3} + 2 Q^{2} x_{1}^{2} x_{2}^{2} x_{3}^{2} + Q^{2} x_{1}^{2} x_{2} x_{3}^{3} + Q^{2} x_{1} x_{2}^{3} x_{3}^{2} + Q^{2} x_{1} x_{2}^{2} x_{3}^{3} - Q x_{1}^{4} x_{2} x_{3} - Q x_{1}^{3} x_{2}^{3} - 3 Q x_{1}^{3} x_{2}^{2} x_{3} - 3 Q x_{1}^{3} x_{2} x_{3}^{2} - Q x_{1}^{3} x_{3}^{3} - 3 Q x_{1}^{2} x_{2}^{3} x_{3} - 4 Q x_{1}^{2} x_{2}^{2} x_{3}^{2} - 3 Q x_{1}^{2} x_{2} x_{3}^{3} - Q x_{1} x_{2}^{4} x_{3} - 3 Q x_{1} x_{2}^{3} x_{3}^{2} - 3 Q x_{1} x_{2}^{2} x_{3}^{3} - Q x_{1} x_{2} x_{3}^{4} - Q x_{2}^{3} x_{3}^{3} + x_{1}^{4} x_{2}^{2} + x_{1}^{4} x_{2} x_{3} + x_{1}^{4} x_{3}^{2} + x_{1}^{3} x_{2}^{3} + 2 x_{1}^{3} x_{2}^{2} x_{3} + 2 x_{1}^{3} x_{2} x_{3}^{2} + x_{1}^{3} x_{3}^{3} + x_{1}^{2} x_{2}^{4} + 2 x_{1}^{2} x_{2}^{3} x_{3} + 3 x_{1}^{2} x_{2}^{2} x_{3}^{2} + 2 x_{1}^{2} x_{2} x_{3}^{3} + x_{1}^{2} x_{3}^{4} + x_{1} x_{2}^{4} x_{3} + 2 x_{1} x_{2}^{3} x_{3}^{2} + 2 x_{1} x_{2}^{2} x_{3}^{3} + x_{1} x_{2} x_{3}^{4} + x_{2}^{4} x_{3}^{2} + x_{2}^{3} x_{3}^{3} + x_{2}^{2} x_{3}^{4}$"
      ],
      "text/plain": [
       "-Q**3*x1**2*x2**2*x3**2 + Q**2*x1**3*x2**2*x3 + Q**2*x1**3*x2*x3**2 + Q**2*x1**2*x2**3*x3 + 2*Q**2*x1**2*x2**2*x3**2 + Q**2*x1**2*x2*x3**3 + Q**2*x1*x2**3*x3**2 + Q**2*x1*x2**2*x3**3 - Q*x1**4*x2*x3 - Q*x1**3*x2**3 - 3*Q*x1**3*x2**2*x3 - 3*Q*x1**3*x2*x3**2 - Q*x1**3*x3**3 - 3*Q*x1**2*x2**3*x3 - 4*Q*x1**2*x2**2*x3**2 - 3*Q*x1**2*x2*x3**3 - Q*x1*x2**4*x3 - 3*Q*x1*x2**3*x3**2 - 3*Q*x1*x2**2*x3**3 - Q*x1*x2*x3**4 - Q*x2**3*x3**3 + x1**4*x2**2 + x1**4*x2*x3 + x1**4*x3**2 + x1**3*x2**3 + 2*x1**3*x2**2*x3 + 2*x1**3*x2*x3**2 + x1**3*x3**3 + x1**2*x2**4 + 2*x1**2*x2**3*x3 + 3*x1**2*x2**2*x3**2 + 2*x1**2*x2*x3**3 + x1**2*x3**4 + x1*x2**4*x3 + 2*x1*x2**3*x3**2 + 2*x1*x2**2*x3**3 + x1*x2*x3**4 + x2**4*x3**2 + x2**3*x3**3 + x2**2*x3**4"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Let us build the coordinates n > len(part)\n",
    "n = 3\n",
    "x = ut.create_x_coord(n)\n",
    "\n",
    "# Explicit expression for the Hall-Littlewood polynomial\n",
    "hl.explicit(x, Q).expand().factor().expand()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "8e37c3dc-4383-4947-9677-5b15e888c3cf",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Let us build the coordinates n < len(part)\n",
    "n = 1\n",
    "x = ut.create_x_coord(n)\n",
    "\n",
    "# Explicit expression for the Hall-Littlewood polynomial\n",
    "hl.explicit(x, Q)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4900aded-7019-4179-b058-be48f5ad7ca8",
   "metadata": {},
   "source": [
    "## Limits Q=0 and Q=1"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "04a29394-5412-49b7-a55f-f6f3c411a5af",
   "metadata": {},
   "source": [
    "One important thing: We need to crete the x coordinates as well the power sums for them, since the Schur polynomials are defined via Miwa coordinates. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "039a2fec-5509-494b-95b6-73c04d0ee50e",
   "metadata": {},
   "outputs": [],
   "source": [
    "# We want to find the limits Q = 0 and Q = 1\n",
    "sc = sy.SchurPolynomial(yg)\n",
    "mp = sy.MonomialPolynomial(yg)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "b80b328b-1e67-4768-ad49-03201b34e21e",
   "metadata": {},
   "outputs": [],
   "source": [
    "n = 2\n",
    "x = ut.create_x_coord(n)\n",
    "X = ut.tx_power_sum(yg.boxes, n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "739e8702-6a2c-4d9b-80ec-4465d5ea3b8c",
   "metadata": {},
   "outputs": [],
   "source": [
    "hl = sy.HallLittlewoodPolynomial(yg)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "094a8766-0d22-457d-a675-bf5db9f979f3",
   "metadata": {},
   "source": [
    "### Schur Polynomials"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "6bf6f92d-8c47-451f-a62d-3a8ce9afb1d2",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# The limit Q=0 is the Schur polynomial.\n",
    "hl.explicit(x, 0).factor() == sc.explicit(X).factor()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "58d57d66-96da-4057-9c3f-e9b72af35901",
   "metadata": {},
   "source": [
    "### Monomial Symmetric Polynomials"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "e73cb1a8-3329-495d-a31d-ae6820a50d36",
   "metadata": {},
   "outputs": [],
   "source": [
    "hl = sy.HallLittlewoodPolynomial(yg)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "60a087e8-cade-40b7-b2d8-1e6934a637c1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# The limit Q=1 is the Schur polynomial.\n",
    "hl.explicit(x, 1).factor() == mp.explicit(x).factor()"
   ]
  }
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