{ "cells": [ { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "# Exploring symmetric function features" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": true, "deletable": true, "editable": true }, "outputs": [], "source": [ "%display latex" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": true, "deletable": true, "editable": true }, "outputs": [], "source": [ "S = SymmetricFunctions(QQ)" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "/opt/sage-git2/local/lib/python2.7/site-packages/sage/combinat/sf/sf.py:1416: RuntimeWarning: redefining global value `e`\n", " inject_variable(shorthand, getattr(self, shorthand)())\n" ] } ], "source": [ "S.inject_shorthands()" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-6s_{1,1,1} - 6s_{1,1,1,1,1,1,1} + 4s_{2,1} - 8s_{2,1,1,1,1,1} - 4s_{2,2,1,1,1} - 8s_{2,2,2,1} + 2s_{3,1,1,1,1} + 6s_{3,2,1,1} - 2s_{3,2,2} + 4s_{3,3,1} + 4s_{4,1,1,1} + 4s_{4,2,1}$ " ], "text/plain": [ "-6*s[1, 1, 1] - 6*s[1, 1, 1, 1, 1, 1, 1] + 4*s[2, 1] - 8*s[2, 1, 1, 1, 1, 1] - 4*s[2, 2, 1, 1, 1] - 8*s[2, 2, 2, 1] + 2*s[3, 1, 1, 1, 1] + 6*s[3, 2, 1, 1] - 2*s[3, 2, 2] + 4*s[3, 3, 1] + 4*s[4, 1, 1, 1] + 4*s[4, 2, 1]" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(s[2,1] + 3 * m[2,1]) * (e[3,1]+1)" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "s_{10,5,4,3,3,2} + s_{10,5,4,4,2,2} + s_{10,5,4,4,3,1} + s_{10,5,5,3,2,2} + s_{10,5,5,3,3,1} + s_{10,5,5,4,2,1} + s_{10,5,5,4,3} + s_{10,6,3,3,3,2} + 2s_{10,6,4,3,2,2} + 2s_{10,6,4,3,3,1} + 2s_{10,6,4,4,2,1} + s_{10,6,4,4,3} + s_{10,6,5,2,2,2} + 3s_{10,6,5,3,2,1} + 2s_{10,6,5,3,3} + s_{10,6,5,4,1,1} + 2s_{10,6,5,4,2} + s_{10,6,6,2,2,1} + s_{10,6,6,3,1,1} + 2s_{10,6,6,3,2} + s_{10,6,6,4,1} + s_{10,7,3,3,2,2} + s_{10,7,3,3,3,1} + s_{10,7,4,2,2,2} + 3s_{10,7,4,3,2,1} + 2s_{10,7,4,3,3} + s_{10,7,4,4,1,1} + 2s_{10,7,4,4,2} + 2s_{10,7,5,2,2,1} + 2s_{10,7,5,3,1,1} + 4s_{10,7,5,3,2} + 2s_{10,7,5,4,1} + s_{10,7,6,2,1,1} + 2s_{10,7,6,2,2} + 3s_{10,7,6,3,1} + s_{10,7,6,4} + s_{10,7,7,2,1} + s_{10,7,7,3} + s_{10,8,3,3,2,1} + s_{10,8,3,3,3} + s_{10,8,4,2,2,1} + s_{10,8,4,3,1,1} + 3s_{10,8,4,3,2} + s_{10,8,4,4,1} + s_{10,8,5,2,1,1} + 2s_{10,8,5,2,2} + 3s_{10,8,5,3,1} + s_{10,8,5,4} + 2s_{10,8,6,2,1} + 2s_{10,8,6,3} + s_{10,8,7,2} + s_{10,9,3,3,2} + s_{10,9,4,2,2} + s_{10,9,4,3,1} + s_{10,9,5,2,1} + s_{10,9,5,3} + s_{10,9,6,2} + s_{11,5,3,3,3,2} + 2s_{11,5,4,3,2,2} + 2s_{11,5,4,3,3,1} + 2s_{11,5,4,4,2,1} + s_{11,5,4,4,3} + s_{11,5,5,2,2,2} + 3s_{11,5,5,3,2,1} + 2s_{11,5,5,3,3} + s_{11,5,5,4,1,1} + 2s_{11,5,5,4,2} + 2s_{11,6,3,3,2,2} + 2s_{11,6,3,3,3,1} + 2s_{11,6,4,2,2,2} + 6s_{11,6,4,3,2,1} + 3s_{11,6,4,3,3} + 2s_{11,6,4,4,1,1} + 3s_{11,6,4,4,2} + 4s_{11,6,5,2,2,1} + 4s_{11,6,5,3,1,1} + 7s_{11,6,5,3,2} + 3s_{11,6,5,4,1} + 2s_{11,6,6,2,1,1} + 3s_{11,6,6,2,2} + 4s_{11,6,6,3,1} + s_{11,6,6,4} + s_{11,7,3,2,2,2} + 3s_{11,7,3,3,2,1} + 2s_{11,7,3,3,3} + 4s_{11,7,4,2,2,1} + 4s_{11,7,4,3,1,1} + 7s_{11,7,4,3,2} + 3s_{11,7,4,4,1} + 4s_{11,7,5,2,1,1} + 6s_{11,7,5,2,2} + 8s_{11,7,5,3,1} + 2s_{11,7,5,4} + s_{11,7,6,1,1,1} + 6s_{11,7,6,2,1} + 4s_{11,7,6,3} + s_{11,7,7,1,1} + 2s_{11,7,7,2} + s_{11,8,3,2,2,1} + s_{11,8,3,3,1,1} + 3s_{11,8,3,3,2} + 2s_{11,8,4,2,1,1} + 4s_{11,8,4,2,2} + 5s_{11,8,4,3,1} + s_{11,8,4,4} + s_{11,8,5,1,1,1} + 6s_{11,8,5,2,1} + 4s_{11,8,5,3} + 2s_{11,8,6,1,1} + 4s_{11,8,6,2} + s_{11,8,7,1} + s_{11,9,3,2,2} + s_{11,9,3,3,1} + 2s_{11,9,4,2,1} + s_{11,9,4,3} + s_{11,9,5,1,1} + 2s_{11,9,5,2} + s_{11,9,6,1} + s_{12,5,3,3,2,2} + s_{12,5,3,3,3,1} + s_{12,5,4,2,2,2} + 3s_{12,5,4,3,2,1} + 2s_{12,5,4,3,3} + s_{12,5,4,4,1,1} + 2s_{12,5,4,4,2} + 2s_{12,5,5,2,2,1} + 2s_{12,5,5,3,1,1} + 4s_{12,5,5,3,2} + 2s_{12,5,5,4,1} + s_{12,6,3,2,2,2} + 3s_{12,6,3,3,2,1} + 2s_{12,6,3,3,3} + 4s_{12,6,4,2,2,1} + 4s_{12,6,4,3,1,1} + 7s_{12,6,4,3,2} + 3s_{12,6,4,4,1} + 4s_{12,6,5,2,1,1} + 6s_{12,6,5,2,2} + 8s_{12,6,5,3,1} + 2s_{12,6,5,4} + s_{12,6,6,1,1,1} + 5s_{12,6,6,2,1} + 3s_{12,6,6,3} + 2s_{12,7,3,2,2,1} + 2s_{12,7,3,3,1,1} + 4s_{12,7,3,3,2} + 4s_{12,7,4,2,1,1} + 6s_{12,7,4,2,2} + 8s_{12,7,4,3,1} + 2s_{12,7,4,4} + 2s_{12,7,5,1,1,1} + 10s_{12,7,5,2,1} + 6s_{12,7,5,3} + 4s_{12,7,6,1,1} + 6s_{12,7,6,2} + 2s_{12,7,7,1} + s_{12,8,3,2,1,1} + 2s_{12,8,3,2,2} + 3s_{12,8,3,3,1} + s_{12,8,4,1,1,1} + 6s_{12,8,4,2,1} + 4s_{12,8,4,3} + 4s_{12,8,5,1,1} + 6s_{12,8,5,2} + 4s_{12,8,6,1} + s_{12,8,7} + s_{12,9,3,2,1} + s_{12,9,3,3} + s_{12,9,4,1,1} + 2s_{12,9,4,2} + 2s_{12,9,5,1} + s_{12,9,6} + s_{13,5,3,3,2,1} + s_{13,5,3,3,3} + s_{13,5,4,2,2,1} + s_{13,5,4,3,1,1} + 3s_{13,5,4,3,2} + s_{13,5,4,4,1} + s_{13,5,5,2,1,1} + 2s_{13,5,5,2,2} + 3s_{13,5,5,3,1} + s_{13,5,5,4} + s_{13,6,3,2,2,1} + s_{13,6,3,3,1,1} + 3s_{13,6,3,3,2} + 2s_{13,6,4,2,1,1} + 4s_{13,6,4,2,2} + 5s_{13,6,4,3,1} + s_{13,6,4,4} + s_{13,6,5,1,1,1} + 6s_{13,6,5,2,1} + 4s_{13,6,5,3} + 2s_{13,6,6,1,1} + 3s_{13,6,6,2} + s_{13,7,3,2,1,1} + 2s_{13,7,3,2,2} + 3s_{13,7,3,3,1} + s_{13,7,4,1,1,1} + 6s_{13,7,4,2,1} + 4s_{13,7,4,3} + 4s_{13,7,5,1,1} + 6s_{13,7,5,2} + 4s_{13,7,6,1} + s_{13,7,7} + 2s_{13,8,3,2,1} + 2s_{13,8,3,3} + 2s_{13,8,4,1,1} + 4s_{13,8,4,2} + 4s_{13,8,5,1} + 2s_{13,8,6} + s_{13,9,3,2} + s_{13,9,4,1} + s_{13,9,5} + s_{14,5,3,3,2} + s_{14,5,4,2,2} + s_{14,5,4,3,1} + s_{14,5,5,2,1} + s_{14,5,5,3} + s_{14,6,3,2,2} + s_{14,6,3,3,1} + 2s_{14,6,4,2,1} + s_{14,6,4,3} + s_{14,6,5,1,1} + 2s_{14,6,5,2} + s_{14,6,6,1} + s_{14,7,3,2,1} + s_{14,7,3,3} + s_{14,7,4,1,1} + 2s_{14,7,4,2} + 2s_{14,7,5,1} + s_{14,7,6} + s_{14,8,3,2} + s_{14,8,4,1} + s_{14,8,5}\n" ] } ], "source": [ "print latex(s[4,3,2] * s[10,5,3])" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}s_{} \\otimes s_{3,2} + s_{1} \\otimes s_{2,2} + s_{1} \\otimes s_{3,1} + s_{1,1} \\otimes s_{2,1} + s_{2} \\otimes s_{2,1} + s_{2} \\otimes s_{3} + s_{2,1} \\otimes s_{1,1} + s_{2,1} \\otimes s_{2} + s_{2,2} \\otimes s_{1} + s_{3} \\otimes s_{2} + s_{3,1} \\otimes s_{1} + s_{3,2} \\otimes s_{}$ " ], "text/plain": [ "s[] # s[3, 2] + s[1] # s[2, 2] + s[1] # s[3, 1] + s[1, 1] # s[2, 1] + s[2] # s[2, 1] + s[2] # s[3] + s[2, 1] # s[1, 1] + s[2, 1] # s[2] + s[2, 2] # s[1] + s[3] # s[2] + s[3, 1] # s[1] + s[3, 2] # s[]" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "s[3,2].coproduct()" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}s_{3,2} \\otimes p_{2,1} + s_{3,2} \\otimes p_{3}$ " ], "text/plain": [ "s[3, 2] # p[2, 1] + s[3, 2] # p[3]" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "tensor([s[3,2], (p[2,1]+p[3])])" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{3}p_{3,3,3} - \\frac{1}{3}p_{9}$ " ], "text/plain": [ "1/3*p[3, 3, 3] - 1/3*p[9]" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "p[3](s[2,1])" ] } ], "metadata": { "kernelspec": { "display_name": "SageMath 8.0.beta6", "language": "", "name": "sagemath" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 2 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython2", "version": "2.7.13" } }, "nbformat": 4, "nbformat_minor": 2 }