Generalization of the Polarisation Formula for Symmetric Bilinear Forms to Symmetric multilinear Forms - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T11:00:03Z http://mathoverflow.net/feeds/question/98714 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/98714/generalization-of-the-polarisation-formula-for-symmetric-bilinear-forms-to-symmet Generalization of the Polarisation Formula for Symmetric Bilinear Forms to Symmetric multilinear Forms Felix Wutschke 2012-06-03T13:25:08Z 2012-07-31T20:43:54Z <p>Hello,</p> <p>Given a symmetric bilinear form $f:V\times V \to K$ , where $V$ is a vectorspace and $K$ is an appropriate field, define the quadratic form $q:V \to K$ as $q(v):= f(v,v)$. </p> <p>The Polarisation Formula states that $f(x,y) = 1/2\big( q(x+y) - q(x) - q(y)\big)$, which is easily proven. </p> <p>This means that any symmetric bilinear form $f:V\times V \to K$ is fully determined by the values $f(v,v)$ for all $v \in V$. </p> <p>I now want to prove the following theorem: Prove that any symmetric $k$-linear form $M:V\times\cdots \times V \to K$ is determined by the values $M[v]^k := M[v,...,v]$ for all $v\in V$. </p> <p>How does that work? </p> http://mathoverflow.net/questions/98714/generalization-of-the-polarisation-formula-for-symmetric-bilinear-forms-to-symmet/98723#98723 Answer by John Wiltshire-Gordon for Generalization of the Polarisation Formula for Symmetric Bilinear Forms to Symmetric multilinear Forms John Wiltshire-Gordon 2012-06-03T16:42:34Z 2012-06-03T16:42:34Z <p>The reason is Schur-Weyl duality. </p> <p>The subspace $W = \langle \forall v \in V \, | \, v \otimes v \otimes v \otimes \cdots \otimes v\rangle$ forms a $GL(V)$ subrepresentation of $\otimes^kV$ if we allow $GL(V)$ to act diagonally on tensors.</p> <p>If we consider the dual action, which is the symmetric group $S_k$ permuting tensor factors, we see that all of the generators of $W$ have the symmetry type of the trivial representation of $S_k$ since they are invariant under these permutations. It follows that the $GL(V)$ subrepresentation $W$ is contained within $\mbox{Sym}^kV \subset \otimes^kV$.</p> <p>However, by Schur-Weyl duality, the symmetric tensors form an irreducible representation of $GL(V)$---the subrepresentation $W$ is either $0$ or all of $\mbox{Sym}^kV$.</p> <p>It isn't $0$, so every symmetric tensor $s$ can be written</p> <p>$$s = \alpha_0 \cdot v_0 \otimes v_0 \otimes v_0 \otimes \cdots \otimes v_0 + \alpha_1 \cdot v_1 \otimes v_1 \otimes v_1 \otimes \cdots \otimes v_1 + \cdots \alpha_l \cdot v_l \otimes v_l \otimes v_l \otimes \cdots \otimes v_l$$</p> <p>for some suitable choice of $v_i$ and $\alpha_i$.</p> <p>In other words, the $k^{th}$ powers of the elements of $V$ span $\mbox{Sym}^k V$. It follows that knowing a symmetric multilinear form on the $k^{th}$ powers is enough to determine the form.</p> http://mathoverflow.net/questions/98714/generalization-of-the-polarisation-formula-for-symmetric-bilinear-forms-to-symmet/98726#98726 Answer by Bazin for Generalization of the Polarisation Formula for Symmetric Bilinear Forms to Symmetric multilinear Forms Bazin 2012-06-03T17:18:35Z 2012-06-04T16:56:40Z <p>You can be completely explicit in this matter. For $T_j$ in a commutative algebra $$T_1T_2\dots T_k=\frac{1}{2^k k!}\sum_{\epsilon_j=\pm 1} \epsilon_1\dots\epsilon_k(\epsilon_1T_1 +\dots+\varepsilon_{k}T_{k})^k.$$ The following lemma in available in the Euclidean case.</p> <p>Lemma. Let $V$ be an Euclidean finite-dimensional vector space, and $A$ a symmetric $k$-multilinear form. We have $\sup_{\Vert T\Vert=1} \vert{A T^k}\vert =\sup_{\Vert{T_j}\Vert=1} \vert{AT_1\dots T_k}\vert.$</p> <p>This lemma is a consequence of the 1928 paper by O.D. Kellogg [MR1544896]. This is not true in the non-Euclidean case where the inequality $$\sup_{\Vert T\Vert=1} \vert{A T^k}\vert \le \sup_{\Vert{T_j}\Vert=1} \vert{AT_1\dots T_k}\vert\le \kappa_k \sup_{\Vert T\Vert=1} \vert{A T^k}\vert,$$ holds true in general with the best constant $\kappa_{k}= k^k/k!.$</p> http://mathoverflow.net/questions/98714/generalization-of-the-polarisation-formula-for-symmetric-bilinear-forms-to-symmet/103634#103634 Answer by Cristi Stoica for Generalization of the Polarisation Formula for Symmetric Bilinear Forms to Symmetric multilinear Forms Cristi Stoica 2012-07-31T20:43:54Z 2012-07-31T20:43:54Z <p>This Wikipedia article contains the <a href="http://en.wikipedia.org/wiki/Polarization_of_an_algebraic_form" rel="nofollow">Polarization of an algebraic form</a> in general.</p>