demo.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Computing the bicharacter of huge spaces of polynomials"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consider the polynomial ring $R=\\mathbb{Q}[X]$ with $r$ sets of $n$ variables. To fix conventions, we think of $X$ as a matrix of variables:\n",
    "\t$$(x_{i,j})_{0\\leq i<r \\atop 0\\leq j<n}$$\n",
    "to which we add an extra row $\\theta = (\\theta_1, \\theta_2, \\dots, \\theta_n)$.\n",
    "We denote by $X_i = (x_{i1}, x_{i2}, \\dots, x_{in})$ the i-th set of variables for $0 \\leq i < r$. Note that the $\\theta$ variables are of degree 0 and don't appear in any operator we define bellow. They are called inert variables. \n",
    "\n",
    "The polynomial ring $R$ is endowed with an action of the symmetric group $S_n$ by diagonal permutation of the columns and of $GL_r$ by diagonal action on the rows. \n",
    "\n",
    "\n",
    "Let $\\gamma$ be a diagram with $n$ cells, note $(a,b)$ the Cartesian coordinates its cells. The diagram $\\gamma$ can be a partition, a composition or an arbitraty diagram.\n",
    "\n",
    "Then the **determinant $V_{\\gamma}$** associated to a diagram $\\gamma$ is given by $V_{\\gamma} = \\det (x_i^a \\theta_i^b)_{1 \\leq i \\leq n \\atop (a,b) \\in \\gamma}$.\n",
    "\n",
    "The **polarisation operator** $P_{u,v}^k$ is given by\n",
    "\t\t$$P_{u,v}^k = \\sum_{i=1}^{n} v_{i} \\partial_{u_{i}}^k$$\n",
    "where $u$ and $v$ are two different sets of variables of $X$.\n",
    "\n",
    "We consider the smallest subspace containing $V_{\\lambda}$ and closed by partial derivation and polarization. We denote this space by $\\mathcal{E}_{\\lambda}$. Our goal is to compute the $GL_r \\times S_n$-character of $\\mathcal{E}_{\\lambda}$.\n",
    "\n",
    "Note that currently the implemented code works only for diagrams of partitions or compositions. For more general diagrams, a quotient needs to be computed. There is a function that can compute the quotient but for mathematical reasons we can't guarantee the rightness of the result. (See section 4)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*Run the following lines to load the code.*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%display latex\n",
    "%runfile character.py"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1. Quick computation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To obtain the complete character of the subpace generated by $V_\\gamma$ and closed by partial derivatives and polarization in the Schur basis, simply use `compute_character`. This function uses by default all implemented optimisations. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}s_{} \\otimes s_{3,1} + s_{1} \\otimes s_{2,1,1} + s_{1} \\otimes s_{2,2} + s_{1,1} \\otimes s_{1,1,1,1} + s_{2} \\otimes s_{2,1,1} + s_{3} \\otimes s_{1,1,1,1}</script></html>"
      ],
      "text/plain": [
       "s[] # s[3, 1] + s[1] # s[2, 1, 1] + s[1] # s[2, 2] + s[1, 1] # s[1, 1, 1, 1] + s[2] # s[2, 1, 1] + s[3] # s[1, 1, 1, 1]"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E_mu(Partition([3,1]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We may want to factorise the character in terms of representations of $S_n$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\left(3, 1\\right), s_{}\\right), \\left(\\left(1, 1, 1, 1\\right), s_{1,1} + s_{3}\\right), \\left(\\left(2, 2\\right), s_{1}\\right), \\left(\\left(2, 1, 1\\right), s_{1} + s_{2}\\right), \\left(\\left(4\\right), 0\\right)\\right]</script></html>"
      ],
      "text/plain": [
       "[((3, 1), s[]),\n",
       " ((1, 1, 1, 1), s[1, 1] + s[3]),\n",
       " ((2, 2), s[1]),\n",
       " ((2, 1, 1), s[1] + s[2]),\n",
       " ((4,), 0)]"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factorize(E_mu(Partition([3,1])))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Or get the latex code of the result."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1 \\otimes s_{2,2} + s_{1} \\otimes s_{2,1,1} + s_{2} \\otimes s_{1,1,1,1} \n"
     ]
    }
   ],
   "source": [
    "print latex_output_character(E_mu(Partition([2,2])))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The function `compute_character` is persistent which means that the results for all the partitions up to the partitions of 5 (except the partition 5 itself) are in memory. Thus when called for those partitions the function read the result in memory instead of computing it again. \n",
    "If you compute characters for other partitions, they will also be added to the memory."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2. A detail example on partition 21"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Vandermonde determinant"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A diagram can be a partition or define as a list of cells given by their cartesian coordinates. Fro example, here is the partition $21$ given as a list of its cells. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\verb|<diagram.Diagram|\\phantom{\\verb!x!}\\verb|instance|\\phantom{\\verb!x!}\\verb|at|\\phantom{\\verb!x!}\\verb|0x7f636c1c2f80>|</script></html>"
      ],
      "text/plain": [
       "<diagram.Diagram instance at 0x7f636c1c2f80>"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diagram = Diagram([(0,0),(0,1),(1,0)])\n",
    "diagram"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " _  \n",
      "|_|_ \n",
      "|_|_|\n"
     ]
    }
   ],
   "source": [
    "diagram.draw()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}</script></html>"
      ],
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diagram.cells() == Partition([2,1]).cells()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can compute the Vandermonde determinant $V_{\\gamma} = \\det (x_i^a \\theta_i^b)_{1 \\leq i \\leq n \\atop (a,b) \\in \\gamma}$ of a partition or a diagram $\\gamma$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}-x_{01} \\theta_{00} + x_{02} \\theta_{00} + x_{00} \\theta_{01} - x_{02} \\theta_{01} - x_{00} \\theta_{02} + x_{01} \\theta_{02}</script></html>"
      ],
      "text/plain": [
       "-x01*theta00 + x02*theta00 + x00*theta01 - x02*theta01 - x00*theta02 + x01*theta02"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v = vandermonde(Partition([2,1]))\n",
    "v"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}-x_{01} \\theta_{00} + x_{02} \\theta_{00} + x_{00} \\theta_{01} - x_{02} \\theta_{01} - x_{00} \\theta_{02} + x_{01} \\theta_{02}</script></html>"
      ],
      "text/plain": [
       "-x01*theta00 + x02*theta00 + x00*theta01 - x02*theta01 - x00*theta02 + x01*theta02"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v = vandermonde(diagram)\n",
    "v"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The space span by partial derivatives"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " Then we can have the subspace $S$ generated by $V_\\gamma$ and its partials derivatives using `Subspace`. It takes in argument two dictionnaries, one for the generators, here $V_\\gamma$, and one for the operators, here the partial derivatives that can be applied to $V_\\gamma$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\left(0\\right) : \\left(\\theta_{01} - \\theta_{02}, \\theta_{00} - \\theta_{02}\\right), \\left(1\\right) : \\left(x_{01} \\theta_{00} - x_{02} \\theta_{00} - x_{00} \\theta_{01} + x_{02} \\theta_{01} + x_{00} \\theta_{02} - x_{01} \\theta_{02}\\right)\\right\\}</script></html>"
      ],
      "text/plain": [
       "{(0,): (theta01 - theta02, theta00 - theta02),\n",
       " (1,): (x01*theta00 - x02*theta00 - x00*theta01 + x02*theta01 + x00*theta02 - x01*theta02,)}"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "generator = {v.multidegree() : [v]}\n",
    "operators = partial_derivatives(v.parent())\n",
    "S = Subspace(generator, operators)\n",
    "S.basis()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Isotypic components"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We now want to project $S$ on the isotypic components of $S_n$. The result will be a dictionnary of subspaces, each one corresponding to one isotypic component associated to the corresponding partition. The function `IsotypicComponent` takes in argument a subspace and the symmetric group order $n$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "{[1, 1, 1]: <subspace.Subspace object at 0x7f636a0af110>, [2, 1]: <subspace.Subspace object at 0x7f636a395910>}\n"
     ]
    }
   ],
   "source": [
    "S_iso = IsotypicComponent(S, 3)\n",
    "print S_iso"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\left(\\left(1\\right), \\left(1, 1, 1\\right)\\right) : \\left(x_{01} \\theta_{00} - x_{02} \\theta_{00} - x_{00} \\theta_{01} + x_{02} \\theta_{01} + x_{00} \\theta_{02} - x_{01} \\theta_{02}\\right)\\right\\}</script></html>"
      ],
      "text/plain": [
       "{((1,),\n",
       "  (1,\n",
       "   1,\n",
       "   1)): (x01*theta00 - x02*theta00 - x00*theta01 + x02*theta01 + x00*theta02 - x01*theta02,)}"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\left(\\left(0\\right), \\left(2, 1\\right)\\right) : \\left(-\\theta_{00} + \\theta_{02}\\right)\\right\\}</script></html>"
      ],
      "text/plain": [
       "{((0,), (2, 1)): (-theta00 + theta02,)}"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "for subspace in S_iso.values():\n",
    "    show(subspace.basis())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that the dictionnary keys contain two tuples, the first one refers to the degree in $x$ of the polynomials and the second one correspond to the type of the isotypic component. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Polarization"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Then we polarize the isotypic component using `PolarizationSpace` but first we need to define a set of operators for example the polarization operators for 2 sets of variables. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "{[1, 1, 1]: <subspace.Subspace object at 0x7f636a0af4d0>, [2, 1]: <subspace.Subspace object at 0x7f636a370b90>}\n"
     ]
    }
   ],
   "source": [
    "polarizators = polarization_operators(2, max_deg=v.degree())\n",
    "S_pol = PolarizedSpace(S_iso, polarizators)\n",
    "print S_pol"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\left(\\left(0, 1\\right), \\left(1, 1, 1\\right)\\right) : \\left(-x_{11} \\theta_{00} + x_{12} \\theta_{00} + x_{10} \\theta_{01} - x_{12} \\theta_{01} - x_{10} \\theta_{02} + x_{11} \\theta_{02}\\right), \\left(\\left(1, 0\\right), \\left(1, 1, 1\\right)\\right) : \\left(x_{01} \\theta_{00} - x_{02} \\theta_{00} - x_{00} \\theta_{01} + x_{02} \\theta_{01} + x_{00} \\theta_{02} - x_{01} \\theta_{02}\\right)\\right\\}</script></html>"
      ],
      "text/plain": [
       "{((0, 1),\n",
       "  (1,\n",
       "   1,\n",
       "   1)): (-x11*theta00 + x12*theta00 + x10*theta01 - x12*theta01 - x10*theta02 + x11*theta02,),\n",
       " ((1, 0),\n",
       "  (1,\n",
       "   1,\n",
       "   1)): (x01*theta00 - x02*theta00 - x00*theta01 + x02*theta01 + x00*theta02 - x01*theta02,)}"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\left(\\left(0, 0\\right), \\left(2, 1\\right)\\right) : \\left(\\theta_{00} - \\theta_{02}\\right)\\right\\}</script></html>"
      ],
      "text/plain": [
       "{((0, 0), (2, 1)): (theta00 - theta02,)}"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "for subspace in S_pol.values():\n",
    "    show(subspace.basis())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "From this polarized isotypic components, we can easily read off the character. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Bicharacter"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}s_{} \\otimes s_{2,1} + s_{1} \\otimes s_{1,1,1}</script></html>"
      ],
      "text/plain": [
       "s[] # s[2, 1] + s[1] # s[1, 1, 1]"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "character(S_pol)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "By default, the character is given in the Schur basis for both sides but a right basis and a left basis can be specified. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}m_{} \\otimes e_{2,1} - m_{} \\otimes e_{3} + m_{1} \\otimes e_{3}</script></html>"
      ],
      "text/plain": [
       "m[] # e[2, 1] - m[] # e[3] + m[1] # e[3]"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "character(S_pol, left_basis=m, right_basis=e)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3. Other possible options"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Polynomial ring of $V_\\gamma$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If nothing is specified, the determinant $V_\\gamma$ is a polynomial oven one set of variables $x$ and inert variables $\\theta$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\verb|Diagonal|\\phantom{\\verb!x!}\\verb|Polynomial|\\phantom{\\verb!x!}\\verb|Ring|\\phantom{\\verb!x!}\\verb|with|\\phantom{\\verb!x!}\\verb|1|\\phantom{\\verb!x!}\\verb|rows|\\phantom{\\verb!x!}\\verb|of|\\phantom{\\verb!x!}\\verb|3|\\phantom{\\verb!x!}\\verb|variables|\\phantom{\\verb!x!}\\verb|and|\\phantom{\\verb!x!}\\verb|inert|\\phantom{\\verb!x!}\\verb|variables|\\phantom{\\verb!x!}\\verb|over|\\phantom{\\verb!x!}\\verb|Rational|\\phantom{\\verb!x!}\\verb|Field|</script></html>"
      ],
      "text/plain": [
       "Diagonal Polynomial Ring with 1 rows of 3 variables and inert variables over Rational Field"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v = vandermonde(Partition([2,1]))\n",
    "v.parent()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "But it is possible to specify a number of sets of variables for the ring in which $V_\\gamma$ lies. At first sight, it change nothing on the determinant itself but now the parent is different. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}-x_{01} \\theta_{00} + x_{02} \\theta_{00} + x_{00} \\theta_{01} - x_{02} \\theta_{01} - x_{00} \\theta_{02} + x_{01} \\theta_{02}</script></html>"
      ],
      "text/plain": [
       "-x01*theta00 + x02*theta00 + x00*theta01 - x02*theta01 - x00*theta02 + x01*theta02"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\verb|Diagonal|\\phantom{\\verb!x!}\\verb|Polynomial|\\phantom{\\verb!x!}\\verb|Ring|\\phantom{\\verb!x!}\\verb|with|\\phantom{\\verb!x!}\\verb|2|\\phantom{\\verb!x!}\\verb|rows|\\phantom{\\verb!x!}\\verb|of|\\phantom{\\verb!x!}\\verb|3|\\phantom{\\verb!x!}\\verb|variables|\\phantom{\\verb!x!}\\verb|and|\\phantom{\\verb!x!}\\verb|inert|\\phantom{\\verb!x!}\\verb|variables|\\phantom{\\verb!x!}\\verb|over|\\phantom{\\verb!x!}\\verb|Rational|\\phantom{\\verb!x!}\\verb|Field|</script></html>"
      ],
      "text/plain": [
       "Diagonal Polynomial Ring with 2 rows of 3 variables and inert variables over Rational Field"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v2 = vandermonde(Partition([2,1]),2)\n",
    "show(v2)\n",
    "v2.parent()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Get a specific isotypic component"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "When projecting on the isotypic component, it is possible to ask for a specific component, for exemple the alternating component in the case of the partition 3. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\left(\\left(3\\right), \\left(1, 1, 1\\right)\\right) : \\left(-x_{00}^{2} x_{01} + x_{00} x_{01}^{2} + x_{00}^{2} x_{02} - x_{01}^{2} x_{02} - x_{00} x_{02}^{2} + x_{01} x_{02}^{2}\\right)\\right\\}</script></html>"
      ],
      "text/plain": [
       "{((3,),\n",
       "  (1,\n",
       "   1,\n",
       "   1)): (-x00^2*x01 + x00*x01^2 + x00^2*x02 - x01^2*x02 - x00*x02^2 + x01*x02^2,)}"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v = vandermonde(Partition([3]))\n",
    "S = Subspace({v.multidegree(): [v]}, partial_derivatives(v.parent()))\n",
    "S_3 = IsotypicComponent(S, Partition([1,1,1]))\n",
    "S_3.basis()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And we can polarize only this component. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\left(\\left(2, 1\\right), \\left(1, 1, 1\\right)\\right) : \\left(-x_{00} x_{01} x_{10} + \\frac{1}{2} x_{01}^{2} x_{10} + x_{00} x_{02} x_{10} - \\frac{1}{2} x_{02}^{2} x_{10} - \\frac{1}{2} x_{00}^{2} x_{11} + x_{00} x_{01} x_{11} - x_{01} x_{02} x_{11} + \\frac{1}{2} x_{02}^{2} x_{11} + \\frac{1}{2} x_{00}^{2} x_{12} - \\frac{1}{2} x_{01}^{2} x_{12} - x_{00} x_{02} x_{12} + x_{01} x_{02} x_{12}\\right), \\left(\\left(1, 1\\right), \\left(1, 1, 1\\right)\\right) : \\left(-x_{01} x_{10} + x_{02} x_{10} + x_{00} x_{11} - x_{02} x_{11} - x_{00} x_{12} + x_{01} x_{12}\\right), \\left(\\left(3, 0\\right), \\left(1, 1, 1\\right)\\right) : \\left(x_{00}^{2} x_{01} - x_{00} x_{01}^{2} - x_{00}^{2} x_{02} + x_{01}^{2} x_{02} + x_{00} x_{02}^{2} - x_{01} x_{02}^{2}\\right), \\left(\\left(1, 2\\right), \\left(1, 1, 1\\right)\\right) : \\left(-\\frac{1}{2} x_{01} x_{10}^{2} + \\frac{1}{2} x_{02} x_{10}^{2} - x_{00} x_{10} x_{11} + x_{01} x_{10} x_{11} + \\frac{1}{2} x_{00} x_{11}^{2} - \\frac{1}{2} x_{02} x_{11}^{2} + x_{00} x_{10} x_{12} - x_{02} x_{10} x_{12} - x_{01} x_{11} x_{12} + x_{02} x_{11} x_{12} - \\frac{1}{2} x_{00} x_{12}^{2} + \\frac{1}{2} x_{01} x_{12}^{2}\\right), \\left(\\left(0, 3\\right), \\left(1, 1, 1\\right)\\right) : \\left(x_{10}^{2} x_{11} - x_{10} x_{11}^{2} - x_{10}^{2} x_{12} + x_{11}^{2} x_{12} + x_{10} x_{12}^{2} - x_{11} x_{12}^{2}\\right)\\right\\}</script></html>"
      ],
      "text/plain": [
       "{((0, 3),\n",
       "  (1,\n",
       "   1,\n",
       "   1)): (x10^2*x11 - x10*x11^2 - x10^2*x12 + x11^2*x12 + x10*x12^2 - x11*x12^2,),\n",
       " ((1, 1),\n",
       "  (1, 1, 1)): (-x01*x10 + x02*x10 + x00*x11 - x02*x11 - x00*x12 + x01*x12,),\n",
       " ((1, 2),\n",
       "  (1,\n",
       "   1,\n",
       "   1)): (-1/2*x01*x10^2 + 1/2*x02*x10^2 - x00*x10*x11 + x01*x10*x11 + 1/2*x00*x11^2 - 1/2*x02*x11^2 + x00*x10*x12 - x02*x10*x12 - x01*x11*x12 + x02*x11*x12 - 1/2*x00*x12^2 + 1/2*x01*x12^2,),\n",
       " ((2, 1),\n",
       "  (1,\n",
       "   1,\n",
       "   1)): (-x00*x01*x10 + 1/2*x01^2*x10 + x00*x02*x10 - 1/2*x02^2*x10 - 1/2*x00^2*x11 + x00*x01*x11 - x01*x02*x11 + 1/2*x02^2*x11 + 1/2*x00^2*x12 - 1/2*x01^2*x12 - x00*x02*x12 + x01*x02*x12,),\n",
       " ((3, 0),\n",
       "  (1,\n",
       "   1,\n",
       "   1)): (x00^2*x01 - x00*x01^2 - x00^2*x02 + x01^2*x02 + x00*x02^2 - x01*x02^2,)}"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "S_3pol = PolarizedSpace(S_3, polarization_operators(2, max_deg=v.degree()))\n",
    "S_3pol.basis()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Other operators"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Other operators are implemented such as Steenrod operators of degree $k>0$ for $r$ sets of variables. The Steenrod operator $S_x^k$ is given by \n",
    "$$S_x^k := \\sum_i x_i \\partial_{x_i}^{k+1}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\left(0, -1\\right) : \\left[\\verb|*.steenrod_op(i=1,|\\phantom{\\verb!x!}\\verb|row_symmetry=None,|\\phantom{\\verb!x!}\\verb|k=2)|\\right], \\left(-1, 0\\right) : \\left[\\verb|*.steenrod_op(i=0,|\\phantom{\\verb!x!}\\verb|row_symmetry=None,|\\phantom{\\verb!x!}\\verb|k=2)|\\right]\\right\\}</script></html>"
      ],
      "text/plain": [
       "{(-1, 0): [*.steenrod_op(i=0, row_symmetry=None, k=2)],\n",
       " (0, -1): [*.steenrod_op(i=1, row_symmetry=None, k=2)]}"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r = 2\n",
    "k = 1\n",
    "steenrod_operators(r, k)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And symmetric derivatives for a given list of degrees. For $r$ sets of variables, let $d=(d_1, d_2, \\dots, d_r$, then the symmetric derivative associated to $d$ is \n",
    "$$D_d := \\sum_i \\partial_{x_{1i}}^{d_1}\\partial_{x_{2i}}^{d_2} \\dots \\partial_{x_{ri}}^{d_r}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\left(0, -2\\right) : \\left[\\verb|*.symmetric_derivative(row_symmetry=None,|\\phantom{\\verb!x!}\\verb|d=(0,|\\phantom{\\verb!x!}\\verb|2))|\\right], \\left(-2, 0\\right) : \\left[\\verb|*.symmetric_derivative(row_symmetry=None,|\\phantom{\\verb!x!}\\verb|d=(2,|\\phantom{\\verb!x!}\\verb|0))|\\right], \\left(-2, -2\\right) : \\left[\\verb|*.symmetric_derivative(row_symmetry=None,|\\phantom{\\verb!x!}\\verb|d=(2,|\\phantom{\\verb!x!}\\verb|2))|\\right], \\left(-1, -1\\right) : \\left[\\verb|*.symmetric_derivative(row_symmetry=None,|\\phantom{\\verb!x!}\\verb|d=(1,|\\phantom{\\verb!x!}\\verb|1))|\\right], \\left(-2, -1\\right) : \\left[\\verb|*.symmetric_derivative(row_symmetry=None,|\\phantom{\\verb!x!}\\verb|d=(2,|\\phantom{\\verb!x!}\\verb|1))|\\right], \\left(-1, -2\\right) : \\left[\\verb|*.symmetric_derivative(row_symmetry=None,|\\phantom{\\verb!x!}\\verb|d=(1,|\\phantom{\\verb!x!}\\verb|2))|\\right], \\left(0, -1\\right) : \\left[\\verb|*.symmetric_derivative(row_symmetry=None,|\\phantom{\\verb!x!}\\verb|d=(0,|\\phantom{\\verb!x!}\\verb|1))|\\right], \\left(-1, 0\\right) : \\left[\\verb|*.symmetric_derivative(row_symmetry=None,|\\phantom{\\verb!x!}\\verb|d=(1,|\\phantom{\\verb!x!}\\verb|0))|\\right]\\right\\}</script></html>"
      ],
      "text/plain": [
       "{(-2, -2): [*.symmetric_derivative(row_symmetry=None, d=(2, 2))],\n",
       " (-2, -1): [*.symmetric_derivative(row_symmetry=None, d=(2, 1))],\n",
       " (-2, 0): [*.symmetric_derivative(row_symmetry=None, d=(2, 0))],\n",
       " (-1, -2): [*.symmetric_derivative(row_symmetry=None, d=(1, 2))],\n",
       " (-1, -1): [*.symmetric_derivative(row_symmetry=None, d=(1, 1))],\n",
       " (-1, 0): [*.symmetric_derivative(row_symmetry=None, d=(1, 0))],\n",
       " (0, -2): [*.symmetric_derivative(row_symmetry=None, d=(0, 2))],\n",
       " (0, -1): [*.symmetric_derivative(row_symmetry=None, d=(0, 1))]}"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d = [(i,j) for i in range(3) for j in range(3) if i+j>0]\n",
    "symmetric_derivatives(d)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "But any dictionnary of operators you would want to try could be used, including mixing diffrent type of operators. The keys of the dictionnary simply have to correspond the multidegress of the operators. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\left(-1, 1\\right) : \\left[\\verb|*.polarization(i1=0,|\\phantom{\\verb!x!}\\verb|row_symmetry=None,|\\phantom{\\verb!x!}\\verb|i2=1,|\\phantom{\\verb!x!}\\verb|d=1)|\\right], \\left(-2, 0\\right) : \\left[\\verb|*.steenrod_op(i=0,|\\phantom{\\verb!x!}\\verb|row_symmetry=None,|\\phantom{\\verb!x!}\\verb|k=3)|\\right], \\left(1, -1\\right) : \\left[\\verb|*.polarization(i1=1,|\\phantom{\\verb!x!}\\verb|row_symmetry=None,|\\phantom{\\verb!x!}\\verb|i2=0,|\\phantom{\\verb!x!}\\verb|d=1)|\\right], \\left(-2, 1\\right) : \\left[\\verb|*.polarization(i1=0,|\\phantom{\\verb!x!}\\verb|row_symmetry=None,|\\phantom{\\verb!x!}\\verb|i2=1,|\\phantom{\\verb!x!}\\verb|d=2)|\\right], \\left(0, -2\\right) : \\left[\\verb|*.steenrod_op(i=1,|\\phantom{\\verb!x!}\\verb|row_symmetry=None,|\\phantom{\\verb!x!}\\verb|k=3)|\\right], \\left(1, -2\\right) : \\left[\\verb|*.polarization(i1=1,|\\phantom{\\verb!x!}\\verb|row_symmetry=None,|\\phantom{\\verb!x!}\\verb|i2=0,|\\phantom{\\verb!x!}\\verb|d=2)|\\right]\\right\\}</script></html>"
      ],
      "text/plain": [
       "{(-2, 0): [*.steenrod_op(i=0, row_symmetry=None, k=3)],\n",
       " (-2, 1): [*.polarization(i1=0, row_symmetry=None, i2=1, d=2)],\n",
       " (-1, 1): [*.polarization(i1=0, row_symmetry=None, i2=1, d=1)],\n",
       " (0, -2): [*.steenrod_op(i=1, row_symmetry=None, k=3)],\n",
       " (1, -2): [*.polarization(i1=1, row_symmetry=None, i2=0, d=2)],\n",
       " (1, -1): [*.polarization(i1=1, row_symmetry=None, i2=0, d=1)]}"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d = merge(polarization_operators(2, max_deg=2), steenrod_operators(2,2))\n",
    "d"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "These operators can be used when computing the subspace spanned by $V_\\gamma$ and some operators or when computing the polarized space. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Change basis for the character"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the function `character`, we can specify bases for $GL_r$ and $S_n$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}m_{} \\otimes s_{2,1} + m_{1} \\otimes s_{1,1,1}</script></html>"
      ],
      "text/plain": [
       "m[] # s[2, 1] + m[1] # s[1, 1, 1]"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v = vandermonde(Partition([2,1]))\n",
    "S_iso = IsotypicComponent(Subspace({v.multidegree(): [v]}, partial_derivatives(v.parent())), 3)\n",
    "S_pol = PolarizedSpace(S_iso, polarization_operators(2, max_deg=v.degree()))\n",
    "character(S_pol, right_basis=s, left_basis=m)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 4. Other example : diagram (0,1,3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The Vandermonde determinant associated to the diagram (0,1,3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " _ _   _\n",
      "|_|_| |_|\n"
     ]
    }
   ],
   "source": [
    "diagram = Diagram([(0,0),(1,0),(3,0)])\n",
    "diagram.draw()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}(x_{01} - x_{02}) \\cdot (-x_{00} + x_{01}) \\cdot (x_{00} - x_{02}) \\cdot (x_{00} + x_{01} + x_{02})</script></html>"
      ],
      "text/plain": [
       "(x01 - x02) * (-x00 + x01) * (x00 - x02) * (x00 + x01 + x02)"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v = vandermonde(diagram)\n",
    "v.factor()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We create the subspace generated by $V_\\gamma$ and its partial derivatives and the Steenrod operators of degree 1 and 2."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "generator = {v.multidegree() : [v]}\n",
    "operators = merge(merge(partial_derivatives(v.parent()), steenrod_operators(1, 1)), steenrod_operators(1, 2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\left(2\\right) : \\left(x_{00}^{2} - x_{02}^{2}, x_{01}^{2} - x_{02}^{2}, -x_{00} x_{01} + x_{00} x_{02}, -x_{00} x_{01} + x_{01} x_{02}\\right), \\left(0\\right) : \\left(1\\right), \\left(3\\right) : \\left(\\frac{1}{3} x_{00}^{3} - \\frac{1}{3} x_{01}^{3} - x_{00} x_{02}^{2} + x_{01} x_{02}^{2}, -\\frac{1}{3} x_{00}^{3} + x_{00} x_{01}^{2} - x_{01}^{2} x_{02} + \\frac{1}{3} x_{02}^{3}, -x_{00}^{2} x_{01} + \\frac{1}{3} x_{01}^{3} + x_{00}^{2} x_{02} - \\frac{1}{3} x_{02}^{3}\\right), \\left(1\\right) : \\left(x_{00}, x_{01}, x_{02}\\right), \\left(4\\right) : \\left(-x_{00}^{3} x_{01} + x_{00} x_{01}^{3} + x_{00}^{3} x_{02} - x_{01}^{3} x_{02} - x_{00} x_{02}^{3} + x_{01} x_{02}^{3}\\right)\\right\\}</script></html>"
      ],
      "text/plain": [
       "{(0,): (1,),\n",
       " (1,): (x00, x01, x02),\n",
       " (2,): (x00^2 - x02^2, x01^2 - x02^2, -x00*x01 + x00*x02, -x00*x01 + x01*x02),\n",
       " (3,): (1/3*x00^3 - 1/3*x01^3 - x00*x02^2 + x01*x02^2,\n",
       "  -1/3*x00^3 + x00*x01^2 - x01^2*x02 + 1/3*x02^3,\n",
       "  -x00^2*x01 + 1/3*x01^3 + x00^2*x02 - 1/3*x02^3),\n",
       " (4,): (-x00^3*x01 + x00*x01^3 + x00^3*x02 - x01^3*x02 - x00*x02^3 + x01*x02^3,)}"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "S = Subspace(generator, operators)\n",
    "S.basis()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can ask for the character before polarization. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}s_{} \\otimes s_{3} + s_{1} \\otimes s_{2,1} + s_{1} \\otimes s_{3} + 2s_{2} \\otimes s_{2,1} + s_{3} \\otimes s_{1,1,1} + s_{3} \\otimes s_{2,1} + s_{4} \\otimes s_{1,1,1}</script></html>"
      ],
      "text/plain": [
       "s[] # s[3] + s[1] # s[2, 1] + s[1] # s[3] + 2*s[2] # s[2, 1] + s[3] # s[1, 1, 1] + s[3] # s[2, 1] + s[4] # s[1, 1, 1]"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "character(IsotypicComponent(S, 3))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Apply $\\sum_i \\partial_{x_i}$ and $\\sum_i x_i \\partial_{x_i}^2$ to $V_{\\gamma}$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}-3 x_{00}^{2} x_{01} + 3 x_{00} x_{01}^{2} + 3 x_{00}^{2} x_{02} - 3 x_{01}^{2} x_{02} - 3 x_{00} x_{02}^{2} + 3 x_{01} x_{02}^{2}</script></html>"
      ],
      "text/plain": [
       "-3*x00^2*x01 + 3*x00*x01^2 + 3*x00^2*x02 - 3*x01^2*x02 - 3*x00*x02^2 + 3*x01*x02^2"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v.symmetric_derivative((1,))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}-6 x_{00}^{2} x_{01} + 6 x_{00} x_{01}^{2} + 6 x_{00}^{2} x_{02} - 6 x_{01}^{2} x_{02} - 6 x_{00} x_{02}^{2} + 6 x_{01} x_{02}^{2}</script></html>"
      ],
      "text/plain": [
       "-6*x00^2*x01 + 6*x00*x01^2 + 6*x00^2*x02 - 6*x01^2*x02 - 6*x00*x02^2 + 6*x01*x02^2"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v.steenrod_op(0, 2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Polarization"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}s_{} \\otimes s_{3} + s_{1} \\otimes s_{2,1} + s_{1} \\otimes s_{3} + s_{1,1} \\otimes s_{1,1,1} + s_{1,1} \\otimes s_{2,1} + 2s_{2} \\otimes s_{2,1} + s_{2,1} \\otimes s_{1,1,1} + s_{3} \\otimes s_{1,1,1} + s_{3} \\otimes s_{2,1} + s_{4} \\otimes s_{1,1,1}</script></html>"
      ],
      "text/plain": [
       "s[] # s[3] + s[1] # s[2, 1] + s[1] # s[3] + s[1, 1] # s[1, 1, 1] + s[1, 1] # s[2, 1] + 2*s[2] # s[2, 1] + s[2, 1] # s[1, 1, 1] + s[3] # s[1, 1, 1] + s[3] # s[2, 1] + s[4] # s[1, 1, 1]"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r=2\n",
    "polarizators = polarization_operators(r, max_deg=v.degree())\n",
    "S_pol = PolarizedSpace(IsotypicComponent(S, 3), polarizators)\n",
    "character(S_pol)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This example is one of the few for which the quotien works properly. We first compute the character of the image of `S_pol` by the symmetric derivatives using the function `Range`. This function takes in argument a Subspace and a list of operators. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}s_{} \\otimes s_{3} + s_{1} \\otimes s_{2,1} + s_{1,1} \\otimes s_{1,1,1} + s_{2} \\otimes s_{2,1} + s_{3} \\otimes s_{1,1,1}</script></html>"
      ],
      "text/plain": [
       "s[] # s[3] + s[1] # s[2, 1] + s[1, 1] # s[1, 1, 1] + s[2] # s[2, 1] + s[3] # s[1, 1, 1]"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "list_degrees = [tuple(k1 if j==i1 else 0 for j in range(r)) for k1 in range(1, 3+1) for i1 in range(0, r)]\n",
    "symmetric_derivatives = symmetric_derivatives(list_degrees)\n",
    "character(Range(S_pol, symmetric_derivatives))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And now the difference of character. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}s_{1} \\otimes s_{3} + s_{1,1} \\otimes s_{2,1} + s_{2} \\otimes s_{2,1} + s_{2,1} \\otimes s_{1,1,1} + s_{3} \\otimes s_{2,1} + s_{4} \\otimes s_{1,1,1}</script></html>"
      ],
      "text/plain": [
       "s[1] # s[3] + s[1, 1] # s[2, 1] + s[2] # s[2, 1] + s[2, 1] # s[1, 1, 1] + s[3] # s[2, 1] + s[4] # s[1, 1, 1]"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "charac = character(S_pol) - character(Range(S_pol, symmetric_derivatives))\n",
    "charac"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Or in terms of monomial symmetric functions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}m_{1} \\otimes s_{3} + 2m_{1,1} \\otimes s_{2,1} + m_{2} \\otimes s_{2,1} + m_{2,1} \\otimes s_{1,1,1} + m_{2,1} \\otimes s_{2,1} + m_{2,2} \\otimes s_{1,1,1} + m_{3} \\otimes s_{2,1} + m_{3,1} \\otimes s_{1,1,1} + m_{4} \\otimes s_{1,1,1}</script></html>"
      ],
      "text/plain": [
       "m[1] # s[3] + 2*m[1, 1] # s[2, 1] + m[2] # s[2, 1] + m[2, 1] # s[1, 1, 1] + m[2, 1] # s[2, 1] + m[2, 2] # s[1, 1, 1] + m[3] # s[2, 1] + m[3, 1] # s[1, 1, 1] + m[4] # s[1, 1, 1]"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "character(S_pol, left_basis=m) - character(Range(S_pol, symmetric_derivatives), left_basis=m)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 5. No quotient for a diagram with holes"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "     _\n",
      " _ _|_|\n",
      "|_|_|_|\n"
     ]
    }
   ],
   "source": [
    "diagram = Diagram([(0,0),(1,0),(2,0),(2,1)])\n",
    "diagram.draw()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}s_{} \\otimes s_{3,1} + s_{1} \\otimes s_{2,1,1} + s_{1} \\otimes s_{2,2} + s_{1} \\otimes s_{3,1} + s_{1,1} \\otimes s_{1,1,1,1} + s_{1,1} \\otimes s_{2,1,1} + s_{1,1} \\otimes s_{2,2} + 2s_{2} \\otimes s_{2,1,1} + s_{2} \\otimes s_{2,2} + s_{2} \\otimes s_{3,1} + s_{2,1} \\otimes s_{1,1,1,1} + s_{2,1} \\otimes s_{2,1,1} + s_{3} \\otimes s_{1,1,1,1} + 2s_{3} \\otimes s_{2,1,1} + s_{3} \\otimes s_{2,2} + s_{3,1} \\otimes s_{1,1,1,1} + s_{4} \\otimes s_{1,1,1,1} + s_{4} \\otimes s_{2,1,1} + s_{5} \\otimes s_{1,1,1,1}</script></html>"
      ],
      "text/plain": [
       "s[] # s[3, 1] + s[1] # s[2, 1, 1] + s[1] # s[2, 2] + s[1] # s[3, 1] + s[1, 1] # s[1, 1, 1, 1] + s[1, 1] # s[2, 1, 1] + s[1, 1] # s[2, 2] + 2*s[2] # s[2, 1, 1] + s[2] # s[2, 2] + s[2] # s[3, 1] + s[2, 1] # s[1, 1, 1, 1] + s[2, 1] # s[2, 1, 1] + s[3] # s[1, 1, 1, 1] + 2*s[3] # s[2, 1, 1] + s[3] # s[2, 2] + s[3, 1] # s[1, 1, 1, 1] + s[4] # s[1, 1, 1, 1] + s[4] # s[2, 1, 1] + s[5] # s[1, 1, 1, 1]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}s_{} \\otimes s_{3,1} + s_{1} \\otimes s_{2,1,1} + s_{1} \\otimes s_{2,2} + s_{1} \\otimes s_{3,1} + s_{1,1} \\otimes s_{1,1,1,1} + s_{1,1} \\otimes s_{2,1,1} + s_{1,1} \\otimes s_{2,2} + 2s_{2} \\otimes s_{2,1,1} + s_{2} \\otimes s_{2,2} + s_{2} \\otimes s_{3,1} + s_{2,1} \\otimes s_{1,1,1,1} + s_{2,1} \\otimes s_{2,1,1} + s_{3} \\otimes s_{1,1,1,1} + 2s_{3} \\otimes s_{2,1,1} + s_{3} \\otimes s_{2,2} + s_{3,1} \\otimes s_{1,1,1,1} + s_{4} \\otimes s_{1,1,1,1} + s_{4} \\otimes s_{2,1,1} + s_{5} \\otimes s_{1,1,1,1}</script></html>"
      ],
      "text/plain": [
       "s[] # s[3, 1] + s[1] # s[2, 1, 1] + s[1] # s[2, 2] + s[1] # s[3, 1] + s[1, 1] # s[1, 1, 1, 1] + s[1, 1] # s[2, 1, 1] + s[1, 1] # s[2, 2] + 2*s[2] # s[2, 1, 1] + s[2] # s[2, 2] + s[2] # s[3, 1] + s[2, 1] # s[1, 1, 1, 1] + s[2, 1] # s[2, 1, 1] + s[3] # s[1, 1, 1, 1] + 2*s[3] # s[2, 1, 1] + s[3] # s[2, 2] + s[3, 1] # s[1, 1, 1, 1] + s[4] # s[1, 1, 1, 1] + s[4] # s[2, 1, 1] + s[5] # s[1, 1, 1, 1]"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v = vandermonde(diagram)\n",
    "n = diagram.size()\n",
    "op = partial_derivatives(v.parent())\n",
    "S_iso = IsotypicComponent(Subspace({v.multidegree(): [v]}, op), n)\n",
    "\n",
    "# with polarization only\n",
    "r = 2\n",
    "op_pol = polarization_operators(r, max_deg=v.degree())\n",
    "S_pol = PolarizedSpace(S_iso, op_pol)\n",
    "show(character(S_pol))\n",
    "\n",
    "# with polarization and lus steenrod of degree 1\n",
    "op_pol = merge(polarization_operators(r), steenrod_operators(r,1))\n",
    "S_pol = PolarizedSpace(S_iso, op_pol)\n",
    "character(S_pol)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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