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Symmetric function
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<h2 id="symmetrization">Symmetrization</h2> <p class="note">Main article: <a href="/facts/Symmetrization/6XbIgHGx">Symmetrization</a></p> <p>Given any function f {\displaystyle f} in n {\displaystyle n} variables with values in an <a href="/facts/Abelian_group/FfWwmx4R">abelian group</a>, a symmetric function can be constructed by summing values of f {\displaystyle f} over all permutations of the arguments. Similarly, an anti-symmetric function can be constructed by summing over <a href="/facts/Even_permutation/ludhCMJz">even permutations</a> and subtracting the sum over <a href="/facts/Odd_permutation/ludhCMJz">odd permutations</a>. These operations are of course not invertible, and could well result in a function that is identically zero for nontrivial functions f . {\displaystyle f.} The only general case where f {\displaystyle f} can be recovered if both its symmetrization and antisymmetrization are known is when n = 2 {\displaystyle n=2} and the abelian group admits a division by 2 (inverse of doubling); then f {\displaystyle f} is equal to half the sum of its symmetrization and its antisymmetrization. </p> <h2 id="examples">Examples</h2> <ul> <li>Consider the <a href="/facts/Real_number/R02gw5Pb">real</a> function f ( x 1 , x 2 , x 3 ) = ( x − x 1 ) ( x − x 2 ) ( x − x 3 ) . {\displaystyle f(x_{1},x_{2},x_{3})=(x-x_{1})(x-x_{2})(x-x_{3}).} By definition, a symmetric function with n {\displaystyle n} variables has the property that f ( x 1 , x 2 , … , x n ) = f ( x 2 , x 1 , … , x n ) = f ( x 3 , x 1 , … , x n , x n − 1 ) , etc. {\displaystyle f(x_{1},x_{2},\ldots ,x_{n})=f(x_{2},x_{1},\ldots ,x_{n})=f(x_{3},x_{1},\ldots ,x_{n},x_{n-1}),\quad {\text{ etc.}}} In general, the function remains the same for every <a href="/facts/Permutation/JSw9ukmv">permutation</a> of its variables. This means that, in this case, ( x − x 1 ) ( x − x 2 ) ( x − x 3 ) = ( x − x 2 ) ( x − x 1 ) ( x − x 3 ) = ( x − x 3 ) ( x − x 1 ) ( x − x 2 ) {\displaystyle (x-x_{1})(x-x_{2})(x-x_{3})=(x-x_{2})(x-x_{1})(x-x_{3})=(x-x_{3})(x-x_{1})(x-x_{2})} and so on, for all permutations of x 1 , x 2 , x 3 . {\displaystyle x_{1},x_{2},x_{3}.} </li> <li>Consider the function f ( x , y ) = x 2 + y 2 − r 2 . {\displaystyle f(x,y)=x^{2}+y^{2}-r^{2}.} If x {\displaystyle x} and y {\displaystyle y} are interchanged the function becomes f ( y , x ) = y 2 + x 2 − r 2 , {\displaystyle f(y,x)=y^{2}+x^{2}-r^{2},} which yields exactly the same results as the original f ( x , y ) . {\displaystyle f(x,y).} </li> <li>Consider now the function f ( x , y ) = a x 2 + b y 2 − r 2 . {\displaystyle f(x,y)=ax^{2}+by^{2}-r^{2}.} If x {\displaystyle x} and y {\displaystyle y} are interchanged, the function becomes f ( y , x ) = a y 2 + b x 2 − r 2 . {\displaystyle f(y,x)=ay^{2}+bx^{2}-r^{2}.} This function is not the same as the original if a ≠ b , {\displaystyle a\neq b,} which makes it non-symmetric. </li> </ul> <h2 id="applications">Applications</h2> <h3>U-statistics</h3> <p class="note">Main article: <a href="/facts/U-statistic/aJM3B2os">U-statistic</a></p> <p>In <a href="/facts/Statistics/DNPsKGYU">statistics</a>, an n {\displaystyle n} -sample statistic (a function in n {\displaystyle n} variables) that is obtained by <a href="/facts/Bootstrapping_(statistics)/zCHuBeIz">bootstrapping</a> symmetrization of a k {\displaystyle k} -sample statistic, yielding a symmetric function in n {\displaystyle n} variables, is called a <a href="/facts/U-statistic/aJM3B2os">U-statistic</a>. Examples include the <a href="/facts/Sample_mean/Ah8VVVDT">sample mean</a> and <a href="/facts/Sample_variance/ULBJKXD1">sample variance</a>. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Alternating_polynomial/uzUEDCFQ">Alternating polynomial</a></li> <li><a href="/facts/Elementary_symmetric_polynomial/9RFXe9b6">Elementary symmetric polynomial</a> – Mathematical function</li> <li><a href="/facts/Even_and_odd_functions/tGHWJqAa">Even and odd functions</a> – Functions such that f(–x) equals f(x) or –f(x)</li> <li><a href="/facts/Exchangeable_random_variables/tHvmhbRy">Exchangeable random variables</a> – Concept in statistics</li> <li><a href="/facts/Quasisymmetric_function/VSh6oTCV">Quasisymmetric function</a></li> <li><a href="/facts/Ring_of_symmetric_functions/SmENv9o4">Ring of symmetric functions</a></li> <li><a href="/facts/Symmetrization/6XbIgHGx">Symmetrization</a> – process that converts any function in n variables to a symmetric function in n variablesPages displaying wikidata descriptions as a fallback</li> <li><a href="/facts/Vandermonde_polynomial/kSJ70Qw5">Vandermonde polynomial</a> – determinant of Vandermonde matrixPages displaying wikidata descriptions as a fallback</li></ul> <ul><li><a href="/facts/F._N._David/WTeeNGfs">F. N. David</a>, <a href="/facts/M._G._Kendall/Vo8Sq4eb">M. G. Kendall</a> & D. E. Barton (1966) <i>Symmetric Function and Allied Tables</i>, <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>.</li> <li>Joseph P. S. Kung, <a href="/facts/Gian-Carlo_Rota/zLSPLVwR">Gian-Carlo Rota</a>, & <a href="/facts/Catherine_Yan/8EPTv4Uh">Catherine H. Yan</a> (2009) <i><a href="/facts/Combinatorics%3a_The_Rota_Way/XAw2H9bc">Combinatorics: The Rota Way</a></i>, §5.1 Symmetric functions, pp 222–5, Cambridge University Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-73794-4.</li></ul>