{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Computing monomial bases 1\n", "\n", "We will see how to compute polytopes whose lattice points parametrize a basis of highest weight modules for Lie algebras of type $A$. The inequalities for these polytopes can be derived from _Dyck paths_ on the set of roots of the Lie algebra. The polytopes are handled with `Polymake`.\n" ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "\u001b[32m\u001b[1m Updating\u001b[22m\u001b[39m git-repo `https://gitlab.com/johannesflake/polybases.jl.git`\n", "\u001b[32m\u001b[1m Resolving\u001b[22m\u001b[39m package versions...\n", "\u001b[32m\u001b[1m No Changes\u001b[22m\u001b[39m to `~/.julia/environments/v1.6/Project.toml`\n", "\u001b[32m\u001b[1m No Changes\u001b[22m\u001b[39m to `~/.julia/environments/v1.6/Manifest.toml`\n" ] } ], "source": [ "using Pkg\n", "Pkg.add(url=\"https://gitlab.com/johannesflake/polybases.jl.git\")\n", "using Oscar, PolyBases" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's start with the $\\mathfrak{sl}_3$-module for the highest weight whose coordinates in terms of the fundamental weights are $(1,2)$. " ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "pm::Matrix\n", "0 0 0\n", "0 0 1\n", "0 0 2\n", "0 1 0\n", "0 1 1\n", "0 1 2\n", "0 2 0\n", "0 2 1\n", "0 3 0\n", "1 0 0\n", "1 0 1\n", "1 0 2\n", "1 1 0\n", "1 1 1\n", "1 2 0\n" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "(15, 3)" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "polytope, pts = PolyBases.DyckPathA.dyck_poly( (1,2,0) );\n", "display(pts)\n", "display(size(pts))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We get $15$ lattice points and the following polytope: " ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "collapsed": true, "jupyter": { "outputs_hidden": true }, "tags": [] }, "outputs": [ { "data": { "image/svg+xml": [ "\n", "\n", "\n", "\tunnamed\n", "\t\n", "\t\t\n", "\t\t\n", "\t\n", "\t\n", "\t\n", "\t\t\n", "\t\t\n", "\t\t\n", "\t\t\n", "\t\n", "\t\n", "\t\t\n", "\t\t0\n", "\t\t\n", "\t\t1\n", "\t\t\n", "\t\t2\n", "\t\t\n", "\t\t3\n", "\t\t\n", "\t\t4\n", "\t\t\n", "\t\t5\n", "\t\t\n", "\t\t6\n", "\t\n", "\t\n", "" ], "text/html": [ "\n", "\n", "\n", "\n", " \n", " \n", " unnamed\n", " \n", " \n", "\n", "
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       "   
\n",
       "       LINEALITY_DIM\n",
       "       
0
\n",
       "   
\n",
       "\n",
       "   
\n",
       "       LINEALITY_SPACE\n",
       "       
\n",
       "   
\n",
       "\n",
       "   
\n",
       "       MONOID_GRADING\n",
       "       
(7) (0 1)
\n",
       "   
\n",
       "\n",
       "   
\n",
       "       N_VERTICES\n",
       "       
42
\n",
       "   
\n",
       "\n",
       "   
\n",
       "       POINTED\n",
       "       
true
\n",
       "   
\n",
       "\n",
       "   
\n",
       "       VERTICES\n",
       "       
1 1 0 0 0 3 0\n",
       "1 1 0 0 0 2 1\n",
       "1 1 0 2 2 1 0\n",
       "1 1 0 3 2 0 0\n",
       "1 1 0 2 2 0 1\n",
       "1 1 0 0 2 0 1\n",
       "1 1 0 0 2 1 0\n",
       "1 1 0 0 2 0 0\n",
       "1 1 2 1 0 0 0\n",
       "1 1 2 0 0 0 1\n",
       "1 1 2 0 0 1 0\n",
       "1 1 2 0 0 0 0\n",
       "1 1 0 0 0 0 0\n",
       "1 1 0 2 0 0 1\n",
       "1 1 0 0 0 0 1\n",
       "1 1 0 3 0 0 0\n",
       "1 0 0 4 0 0 0\n",
       "1 0 0 0 0 0 1\n",
       "1 0 0 3 0 0 1\n",
       "1 0 0 0 0 3 0\n",
       "1 0 0 1 0 3 0\n",
       "1 0 1 0 0 3 0\n",
       "1 0 0 0 0 2 1\n",
       "1 0 0 1 0 2 1\n",
       "1 0 1 0 0 2 1\n",
       "1 0 0 3 2 1 0\n",
       "1 0 1 2 2 1 0\n",
       "1 0 0 4 2 0 0\n",
       "1 0 1 3 2 0 0\n",
       "1 0 0 3 2 0 1\n",
       "1 0 1 2 2 0 1\n",
       "1 0 0 0 2 0 1\n",
       "1 0 1 0 2 0 1\n",
       "1 0 0 0 2 1 0\n",
       "1 0 1 0 2 1 0\n",
       "1 0 0 0 2 0 0\n",
       "1 0 1 0 2 0 0\n",
       "1 0 3 1 0 0 0\n",
       "1 0 3 0 0 0 1\n",
       "1 0 3 0 0 1 0\n",
       "1 0 3 0 0 0 0\n",
       "1 0 0 0 0 0 0
\n",
       "   
\n",
       "\n",
       "   
\n",
       "       VERTICES_IN_FACETS\n",
       "       
{16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41}\n",
       "{0 1 2 3 4 5 6 7 12 13 14 15 16 17 18 19 20 22 23 25 27 29 31 33 35 41}\n",
       "{0 1 5 6 7 9 10 11 12 14 17 19 21 22 24 31 32 33 34 35 36 38 39 40 41}\n",
       "{0 1 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 37 38 39 40 41}\n",
       "{3 4 5 7 8 9 11 12 13 14 15 16 17 18 27 28 29 30 31 32 35 36 37 38 40 41}\n",
       "{0 2 3 6 7 8 10 11 12 15 16 19 20 21 25 26 27 28 33 34 35 36 37 39 40 41}\n",
       "{0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15}\n",
       "{2 3 4 5 6 7 25 26 27 28 29 30 31 32 33 34 35 36}\n",
       "{1 4 5 9 13 14 17 18 22 23 24 29 30 31 32 38}\n",
       "{2 3 4 5 6 7 8 9 10 11 26 28 30 32 34 36 37 38 39 40}\n",
       "{0 1 2 4 5 6 19 20 21 22 23 24 25 26 29 30 31 32 33 34}\n",
       "{0 1 2 3 4 8 9 10 13 15 16 18 20 21 23 24 25 26 27 28 29 30 37 38 39}\n",
       "{0 1 2 4 5 6 9 10 21 24 26 30 32 34 38 39}
\n",
       "   
\n",
       "\n",
       "   
\n",
       "       VERTICES_IN_INEQUALITIES\n",
       "       
{16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41}\n",
       "{0 1 2 3 4 5 6 7 12 13 14 15 16 17 18 19 20 22 23 25 27 29 31 33 35 41}\n",
       "{0 1 5 6 7 9 10 11 12 14 17 19 21 22 24 31 32 33 34 35 36 38 39 40 41}\n",
       "{0 1 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 37 38 39 40 41}\n",
       "{3 4 5 7 8 9 11 12 13 14 15 16 17 18 27 28 29 30 31 32 35 36 37 38 40 41}\n",
       "{0 2 3 6 7 8 10 11 12 15 16 19 20 21 25 26 27 28 33 34 35 36 37 39 40 41}\n",
       "{0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15}\n",
       "{2 3 4 5 6 7 25 26 27 28 29 30 31 32 33 34 35 36}\n",
       "{1 4 5 9 13 14 17 18 22 23 24 29 30 31 32 38}\n",
       "{2 3 4 5 6 7 8 9 10 11 26 28 30 32 34 36 37 38 39 40}\n",
       "{0 1 2 4 5 6 19 20 21 22 23 24 25 26 29 30 31 32 33 34}\n",
       "{0 1 2 3 4 8 9 10 13 15 16 18 20 21 23 24 25 26 27 28 29 30 37 38 39}\n",
       "{0 1 2 4 5 6 9 10 21 24 26 30 32 34 38 39}\n",
       "{}
\n",
       "   
\n",
       "\n",
       "   
\n",
       "       VERTICES_IN_INEQUALITIES\n",
       "       
{16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41}\n",
       "{0 1 2 3 4 5 6 7 12 13 14 15 16 17 18 19 20 22 23 25 27 29 31 33 35 41}\n",
       "{0 1 5 6 7 9 10 11 12 14 17 19 21 22 24 31 32 33 34 35 36 38 39 40 41}\n",
       "{0 1 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 37 38 39 40 41}\n",
       "{3 4 5 7 8 9 11 12 13 14 15 16 17 18 27 28 29 30 31 32 35 36 37 38 40 41}\n",
       "{0 2 3 6 7 8 10 11 12 15 16 19 20 21 25 26 27 28 33 34 35 36 37 39 40 41}\n",
       "{0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15}\n",
       "{2 3 4 5 6 7 25 26 27 28 29 30 31 32 33 34 35 36}\n",
       "{1 4 5 9 13 14 17 18 22 23 24 29 30 31 32 38}\n",
       "{2 3 4 5 6 7 8 9 10 11 26 28 30 32 34 36 37 38 39 40}\n",
       "{0 1 2 4 5 6 19 20 21 22 23 24 25 26 29 30 31 32 33 34}\n",
       "{0 1 2 3 4 8 9 10 13 15 16 18 20 21 23 24 25 26 27 28 29 30 37 38 39}\n",
       "{0 1 2 4 5 6 9 10 21 24 26 30 32 34 38 39}\n",
       "{}
\n",
       "   
\n"
      ],
      "text/plain": [
       "type: Polytope\n",
       "\n",
       "AFFINE_HULL\n",
       "\t\n",
       "\n",
       "BOUNDARY_LATTICE_POINTS\n",
       "\t1 0 0 0 0 0 0\n",
       "\t1 0 0 0 0 0 1\n",
       "\t1 0 0 0 0 1 0\n",
       "\t1 0 0 0 0 1 1\n",
       "\t1 0 0 0 0 2 0\n",
       "\t1 0 0 0 0 2 1\n",
       "\t1 0 0 0 0 3 0\n",
       "\t1 0 0 0 1 0 0\n",
       "\t1 0 0 0 1 0 1\n",
       "\t1 0 0 0 1 1 0\n",
       "\t…\n",
       "\n",
       "BOUNDED\n",
       "\ttrue\n",
       "\n",
       "COMBINATORIAL_DIM\n",
       "\t6\n",
       "\n",
       "CONE_AMBIENT_DIM\n",
       "\t7\n",
       "\n",
       "CONE_DIM\n",
       "\t7\n",
       "\n",
       "FACETS\n",
       "  0   1   0   0   0   0   0\n",
       "  0   0   1   0   0   0   0\n",
       "  0   0   0   1   0   0   0\n",
       "  0   0   0   0   1   0   0\n",
       "  0   0   0   0   0   1   0\n",
       "  0   0   0   0   0   0   1\n",
       "  1  -1   0   0   0   0   0\n",
       "  2   0   0   0  -1   0   0\n",
       "  1   0   0   0   0   0  -1\n",
       "  3  -1  -1   0  -1   0   0\n",
       "  3   0   0   0  -1  -1  -1\n",
       "  4  -1  -1  -1   0  -1  -1\n",
       "  4  -1  -1   0  -1  -1  -1\n",
       "\n",
       "FEASIBLE\n",
       "\ttrue\n",
       "\n",
       "FULL_DIM\n",
       "\ttrue\n",
       "\n",
       "INEQUALITIES\n",
       "  0   1   0   0   0   0   0\n",
       "  0   0   1   0   0   0   0\n",
       "  0   0   0   1   0   0   0\n",
       "  0   0   0   0   1   0   0\n",
       "  0   0   0   0   0   1   0\n",
       "  0   0   0   0   0   0   1\n",
       "  1  -1   0   0   0   0   0\n",
       "  2   0   0   0  -1   0   0\n",
       "  1   0   0   0   0   0  -1\n",
       "  3  -1  -1   0  -1   0   0\n",
       "  3   0   0   0  -1  -1  -1\n",
       "  4  -1  -1  -1   0  -1  -1\n",
       "  4  -1  -1   0  -1  -1  -1\n",
       "  1   0   0   0   0   0   0\n",
       "\n",
       "INTERIOR_LATTICE_POINTS\n",
       "\t\n",
       "\n",
       "LATTICE_POINTS_GENERATORS\n",
       "\t<1 0 0 0 0 0 0\n",
       "\t1 0 0 0 0 0 1\n",
       "\t1 0 0 0 0 1 0\n",
       "\t1 0 0 0 0 1 1\n",
       "\t1 0 0 0 0 2 0\n",
       "\t1 0 0 0 0 2 1\n",
       "\t1 0 0 0 0 3 0\n",
       "\t1 0 0 0 1 0 0\n",
       "\t1 0 0 0 1 0 1\n",
       "\t1 0 0 0 1 1 0\n",
       "\t…\n",
       "\n",
       "LINEALITY_DIM\n",
       "\t0\n",
       "\n",
       "LINEALITY_SPACE\n",
       "\t\n",
       "\n",
       "MONOID_GRADING\n",
       "\t(7) (0 1)\n",
       "\n",
       "N_VERTICES\n",
       "\t42\n",
       "\n",
       "POINTED\n",
       "\ttrue\n",
       "\n",
       "VERTICES\n",
       "  1  1  0  0  0  3  0\n",
       "  1  1  0  0  0  2  1\n",
       "  1  1  0  2  2  1  0\n",
       "  1  1  0  3  2  0  0\n",
       "  1  1  0  2  2  0  1\n",
       "  1  1  0  0  2  0  1\n",
       "  1  1  0  0  2  1  0\n",
       "  1  1  0  0  2  0  0\n",
       "  1  1  2  1  0  0  0\n",
       "  1  1  2  0  0  0  1\n",
       "  1  1  2  0  0  1  0\n",
       "  1  1  2  0  0  0  0\n",
       "  1  1  0  0  0  0  0\n",
       "  ⋮              ⋮  \n",
       "  1  0  1  2  2  0  1\n",
       "  1  0  0  0  2  0  1\n",
       "  1  0  1  0  2  0  1\n",
       "  1  0  0  0  2  1  0\n",
       "  1  0  1  0  2  1  0\n",
       "  1  0  0  0  2  0  0\n",
       "  1  0  1  0  2  0  0\n",
       "  1  0  3  1  0  0  0\n",
       "  1  0  3  0  0  0  1\n",
       "  1  0  3  0  0  1  0\n",
       "  1  0  3  0  0  0  0\n",
       "  1  0  0  0  0  0  0\n",
       "\n",
       "VERTICES_IN_FACETS\n",
       "\t{16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41}\n",
       "\t{0 1 2 3 4 5 6 7 12 13 14 15 16 17 18 19 20 22 23 25 27 29 31 33 35 41}\n",
       "\t{0 1 5 6 7 9 10 11 12 14 17 19 21 22 24 31 32 33 34 35 36 38 39 40 41}\n",
       "\t{0 1 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 37 38 39 40 41}\n",
       "\t{3 4 5 7 8 9 11 12 13 14 15 16 17 18 27 28 29 30 31 32 35 36 37 38 40 41}\n",
       "\t{0 2 3 6 7 8 10 11 12 15 16 19 20 21 25 26 27 28 33 34 35 36 37 39 40 41}\n",
       "\t{0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15}\n",
       "\t{2 3 4 5 6 7 25 26 27 28 29 30 31 32 33 34 35 36}\n",
       "\t{1 4 5 9 13 14 17 18 22 23 24 29 30 31 32 38}\n",
       "\t{2 3 4 5 6 7 8 9 10 11 26 28 30 32 34 36 37 38 39 40}\n",
       "\t…\n",
       "\n",
       "VERTICES_IN_INEQUALITIES\n",
       "\t{16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41}\n",
       "\t{0 1 2 3 4 5 6 7 12 13 14 15 16 17 18 19 20 22 23 25 27 29 31 33 35 41}\n",
       "\t{0 1 5 6 7 9 10 11 12 14 17 19 21 22 24 31 32 33 34 35 36 38 39 40 41}\n",
       "\t{0 1 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 37 38 39 40 41}\n",
       "\t{3 4 5 7 8 9 11 12 13 14 15 16 17 18 27 28 29 30 31 32 35 36 37 38 40 41}\n",
       "\t{0 2 3 6 7 8 10 11 12 15 16 19 20 21 25 26 27 28 33 34 35 36 37 39 40 41}\n",
       "\t{0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15}\n",
       "\t{2 3 4 5 6 7 25 26 27 28 29 30 31 32 33 34 35 36}\n",
       "\t{1 4 5 9 13 14 17 18 22 23 24 29 30 31 32 38}\n",
       "\t{2 3 4 5 6 7 8 9 10 11 26 28 30 32 34 36 37 38 39 40}\n",
       "\t…\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "polytope2, pts2 = PolyBases.DyckPathA.dyck_poly( (1,2,1,0) );\n",
    "display(size(pts2, 1))\n",
    "display(polytope2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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   "language": "julia",
   "name": "julia-1.6"
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  "language_info": {
   "file_extension": ".jl",
   "mimetype": "application/julia",
   "name": "julia",
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 "nbformat": 4,
 "nbformat_minor": 4
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