{
 "cells": [
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   "execution_count": 1,
   "id": "ae32a5a2-a3e8-422e-b3c7-a81d220eda9d",
   "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": "c7dfe4ed-72d2-40a3-a701-93f3e49a1cf0",
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    "jp-MarkdownHeadingCollapsed": true
   },
   "source": [
    "# Class: SchurPolynomial"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ac8ef389-f1ff-4796-a32c-60e1c601dd63",
   "metadata": {},
   "source": [
    "Let us now consider the Schur polynomials. We can initialize the SchurPolynomials class as: sy.SchurPolynomial(young: YoungDiagram)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "88cdb597-c278-470b-87ba-3c820fc040a8",
   "metadata": {},
   "outputs": [],
   "source": [
    "par = (3, 2, 1) # define a partition\n",
    "yg = sy.YoungDiagram(par) # Initialize an object, YoungDiagram\n",
    "sch = sy.SchurPolynomial(yg) # Initialize the Schur polynomial"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "6fd4f0e6-909c-48a1-b8f3-d45a66f3f23d",
   "metadata": {},
   "outputs": [],
   "source": [
    "t = ut.create_miwa(yg.boxes) # using the function in the utils module\n",
    "tt = tuple(t.values())"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f969c391-2be1-4b05-84c6-178417e30b16",
   "metadata": {},
   "source": [
    "To get the explicit expression for the polynomial, one can use the method .explicit(t). As before, it accepts tuples and dictionaries. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "ed183a61-8b95-4102-b538-9b3c543e1c95",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{t_{1}^{6}}{45} - \\frac{t_{1}^{3} t_{3}}{3} + t_{1} t_{5} - t_{3}^{2}$"
      ],
      "text/plain": [
       "t1**6/45 - t1**3*t3/3 + t1*t5 - t3**2"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sch.explicit(t)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1cbd112d-d40e-4bdc-a5b1-82ce9528ca9b",
   "metadata": {},
   "source": [
    "As before, one can get the sympy polynomials as"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "095b3e05-ee98-4b78-89e8-1c878a73af88",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\operatorname{Poly}{\\left( \\frac{1}{45} t_{1}^{6} -  \\frac{1}{3} t_{1}^{3}t_{3} + t_{1}t_{5} - t_{3}^{2}, t_{1}, t_{3}, t_{5}, domain=\\mathbb{Q} \\right)}$"
      ],
      "text/plain": [
       "Poly(1/45*t1**6 - 1/3*t1**3*t3 + t1*t5 - t3**2, t1, t3, t5, domain='QQ')"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sch.explicit(tt, True)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f2b497e3-95b6-4cfb-bf83-4689a2f6be86",
   "metadata": {},
   "source": [
    "One can also express the Schur polynomials in terms of the coordinates x:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "bbc1e73f-d021-485a-9f0f-868c1f7d4bdb",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(x1 + x2 + x3,\n",
       " x1**2/2 + x2**2/2 + x3**2/2,\n",
       " x1**3/3 + x2**3/3 + x3**3/3,\n",
       " x1**4/4 + x2**4/4 + x3**4/4,\n",
       " x1**5/5 + x2**5/5 + x3**5/5,\n",
       " x1**6/6 + x2**6/6 + x3**6/6)"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Suppose we have x = (x1, x2, x3), \n",
    "m = 3 # minimum to give a non trivial \n",
    "tx = ut.tx_power_sum(yg.boxes, m)\n",
    "tx"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "8d137942-6ed8-4475-ba45-c08cc0b55f87",
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    {
     "data": {
      "text/latex": [
       "$\\displaystyle x_{1}^{3} x_{2}^{2} x_{3} + x_{1}^{3} x_{2} x_{3}^{2} + x_{1}^{2} x_{2}^{3} x_{3} + 2 x_{1}^{2} x_{2}^{2} x_{3}^{2} + x_{1}^{2} x_{2} x_{3}^{3} + x_{1} x_{2}^{3} x_{3}^{2} + x_{1} x_{2}^{2} x_{3}^{3}$"
      ],
      "text/plain": [
       "x1**3*x2**2*x3 + x1**3*x2*x3**2 + x1**2*x2**3*x3 + 2*x1**2*x2**2*x3**2 + x1**2*x2*x3**3 + x1*x2**3*x3**2 + x1*x2**2*x3**3"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sch.explicit(tx)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a95cdc21-0dc5-4b96-acf6-e4998c9f4450",
   "metadata": {
    "jp-MarkdownHeadingCollapsed": true
   },
   "source": [
    "## Generate Schur polynomials for a given level n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "58772de8-355d-4fa6-9866-9abf42f9c0ac",
   "metadata": {},
   "source": [
    "One can easily generate all Schur polynomials for a given level n as follows:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "id": "103ea948-f04f-4fb0-8fa4-08389bbcf5ba",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Define a function to list the polynomials\n",
    "def list_schur(n):\n",
    "    states = sy.State(n).partition_states()\n",
    "    for a in states:\n",
    "        yg = sy.YoungDiagram(a)\n",
    "        sch = sy.SchurPolynomial(yg)\n",
    "        t = ut.create_miwa(yg.boxes)\n",
    "        print(f'-- Schur[{a}] --')\n",
    "        display(sch.explicit(t))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "id": "b763fbce-9253-408f-95d6-df2a309946c3",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-- Schur[(1, 1)] --\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{t_{1}^{2}}{2} - t_{2}$"
      ],
      "text/plain": [
       "t1**2/2 - t2"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-- Schur[(2, 0)] --\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{t_{1}^{2}}{2} + t_{2}$"
      ],
      "text/plain": [
       "t1**2/2 + t2"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "list_schur(2)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "92968a26-1f97-43b8-87a9-ce642bdd641e",
   "metadata": {},
   "source": [
    "## Skew-Schur Polynomials"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ab462a1c-69d6-4bd5-96b2-49b6f3388e9c",
   "metadata": {},
   "source": [
    "In order to define the Schur polynomials, we need to consider a secong Young diagram $\\mu$ that is contained in $\\lambda$, we write $\\mu \\subseteq \\lambda$. We can use the method constains(mu) to test the subset relation between the partitions. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "id": "a86ae362-923a-466f-ae7e-a0d27cf625df",
   "metadata": {},
   "outputs": [],
   "source": [
    "yg1 = sy.YoungDiagram((4,3,2,2,1))\n",
    "yg2 = sy.YoungDiagram((3,2,1))\n",
    "t = ut.create_miwa(yg1.boxes)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "id": "5f3b0a44-7ff3-4494-8563-2bc3e590f647",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 86,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "yg1.contains(yg2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "id": "c12d627d-1ab0-44f4-b747-7e4c1416c01c",
   "metadata": {},
   "outputs": [],
   "source": [
    "# We find skew Schur polynomials with the skew_schur method\n",
    "sch1 = sy.SchurPolynomial(yg1)\n",
    "sch2 = sy.SchurPolynomial(yg2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "id": "950600f6-1e95-455e-a046-a51f9d3b8792",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 88,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sch.skew_schur(t, yg1) # bacause yg2 does not contain yg1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "id": "da9b9288-0172-42b9-bbab-82f3f8719160",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle t_{1}^{2} \\cdot \\left(\\frac{5 t_{1}^{4}}{24} - \\frac{t_{1}^{2} t_{2}}{2} - t_{1} t_{3} + \\frac{t_{2}^{2}}{2} + t_{4}\\right)$"
      ],
      "text/plain": [
       "t1**2*(5*t1**4/24 - t1**2*t2/2 - t1*t3 + t2**2/2 + t4)"
      ]
     },
     "execution_count": 89,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sch1.skew_schur(t, yg2) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "e63b4115-c3be-4553-b39f-d5d60a5114f4",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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