130 angles <math><latex>\theta</latex></math> et |
130 angles <math><latex>\theta</latex></math> et |
131 <math><latex>\varphi</latex></math> de sorte que toute fonction sur la |
131 <math><latex>\varphi</latex></math> de sorte que toute fonction sur la |
132 sphère peut se décomposer sur une base d'harmoniques sphériques. |
132 sphère peut se décomposer sur une base d'harmoniques sphériques. |
133 </p> |
133 </p> |
134 </section> |
134 </section> |
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135 |
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136 <section> |
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137 <head> |
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138 <title>Formule avec préambule</title> |
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139 </head> |
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140 <section xml:lang="en"> |
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141 <p> |
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142 Let <math><latex>I_+</latex></math> denote the ideal generated by |
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143 the <math><latex>S_n</latex></math>-invariant homogeneous |
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144 polynomials of positive degree in <math> |
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145 <preambule><newcommand name="C">{\mathbb{C}}</newcommand></preambule> |
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146 <latex>\C[x_1,\dots, x_n,y_1,\dots, y_n]</latex></math> and set |
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147 </p> |
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148 <p> |
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149 <math display="wide"> |
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150 <preambule><newcommand name="C">{\mathbb{C}}</newcommand></preambule> |
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151 <latex>R_n :=\C[\mathbf{x},\mathbf{y}] / I_+.</latex> |
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152 </math> |
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153 </p> |
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154 <p> |
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155 It is known for sometime that the bi-graded Frobenius character of |
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156 <math><latex>R_n</latex></math> is given by the transformation of |
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157 the elementary symmetric function <math><latex>e_{n}</latex></math> |
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158 under Bergeron-Garsia's "nabla" operator, <math><latex>\nabla |
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159 e_n</latex></math>. In other words, the symmetric function |
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160 <math><latex>\nabla e_n</latex></math> has an underlying |
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161 <math><latex>S_{n}</latex></math>-representation. Roughly stated, |
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162 <math><latex>\nabla</latex></math> is a |
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163 <math> |
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164 <preambule> |
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165 <newcommand name="mbb">[1]{\mathbb{#1}}</newcommand></preambule> |
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166 <latex>\mbb{Q}</latex></math>-linear operator defined on the ring |
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167 of symmetric functions <math><latex>\Lambda</latex></math> in |
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168 such a way that the modified Macdonald symmetric functions are |
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169 the eigenfunctions of <math><latex>\nabla</latex></math> with |
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170 prescribed eigenvalues. |
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171 </p> |
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172 </section> |
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173 </section> |
135 </section> |
174 </section> |
136 </topic> |
175 </topic> |
137 </publidoc> |
176 </publidoc> |