Умножение симметрий и преобразований

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<ul><li><p> ..</p><p>115</p><p>. . </p><p> . .. </p><p>1. </p><p> -, -, - , - - . -, , - , - , , , .</p><p>, , - .</p><p> ( ) - - - , , , , . </p><p> , - , - , . - -. - . , , .</p><p>2. </p><p> - ., , - , -. , </p><p>AB</p><p>CD</p><p> (. 1</p><p>), - </p><p>1</p><p>3</p><p>2</p><p>4</p><p>MULTIPLICATIONOF SYMMETRIESAND TRANSFORMATIONS</p><p>A. Yu. OLSHANSKII</p><p>This paper introducesthe group theoreticalpoint of view of symme-tries and transformations.Symmetries of polynomi-als are used as a defini-tion for permutationssign, which is appliedafterwards to a puzzle.Multiplication of symme-tries is quite different fromusual numerical opera-tions. Therefore a cal-culus arises which leadsto general notions of thegroup theory.</p><p> -- - - . -, . - - - -. -, - .</p><p>' </p><p>.., </p><p>1996</p></li><li><p> , 5, 1996</p><p>116</p><p> 0</p><p>, 90</p><p>, 180</p><p> 270</p><p> .</p><p> 8 0</p><p>(- , -). , , . 360</p><p> - , 90</p><p> - 450</p><p>: , - .</p><p> (. 1</p><p>) - 3 : 180</p><p>AB</p><p>CD</p><p>. (. 1</p><p>) - . -, . </p><p> ( - ) , , (. 2</p><p>) ( ), - (. 2</p><p>), -</p><p> , </p><p>k</p><p>-</p><p>l</p><p>k</p><p>l</p><p> 3.</p><p>xy</p><p> + </p><p>xz </p><p>+ </p><p>yz</p><p> , </p><p>xy</p><p>2</p><p> + </p><p>yz</p><p>2</p><p> + </p><p>zx</p><p>2</p><p>, -</p><p>3a 3b 6a 6b , , , ,ka lb+</p><p> , </p><p>x</p><p> + 2</p><p>y</p><p> + 5</p><p>z</p><p>. -, , - , - </p><p>x</p><p>, </p><p>y</p><p>z</p><p> . , , , </p><p>x</p><p>y</p><p>, </p><p>y</p><p>z</p><p>, </p><p>z</p><p>x</p><p>. -</p><p> 1. -, - 1 (!), - , - .</p><p> 1</p><p> , , .. -, </p><p> . , 6, - 8. , , .</p><p>3. </p><p> - (, , - ).</p><p>f </p><p>- </p><p>X</p><p>?</p><p>X </p><p> - </p><p>. ( - . !), </p><p>( - , , ) </p><p>f </p><p>X</p><p>, </p><p>a</p><p>X </p><p>(</p><p>a</p><p>X</p><p>) </p><p>X</p><p>, - </p><p>a</p><p>f</p><p>(</p><p>a</p><p>); , </p><p>b </p><p>X </p><p>(</p><p>b</p><p>X</p><p>) , , </p><p>a</p><p>X</p><p>. </p><p>a </p><p>b.</p><p> . </p><p>x y z</p><p>y x z</p><p>x y z</p><p>z y x</p><p>x y z</p><p>x z y</p><p>x y z</p><p>y z x</p><p>x y z</p><p>z x y</p><p>x y z</p><p>x y z</p><p>. 1.</p><p>. 2.</p><p>1</p><p>4 3</p><p>2</p><p>A B</p><p>C</p><p>D</p><p>A B</p><p>C</p><p>D</p><p>a b</p></li><li><p> ..</p><p>117</p><p>1. </p><p>X </p><p> , </p><p>f </p><p> - 90</p><p> . (1, 0) (0, </p><p>-</p><p>1), (0, </p><p>-</p><p>1) (</p><p>-</p><p>1, 0). , - (</p><p>x, y</p><p>) (</p><p>y</p><p>, </p><p>-</p><p>x</p><p>), (</p><p>x</p><p>,</p><p> y</p><p>) (</p><p>-</p><p>y</p><p>, </p><p>x</p><p>).2. </p><p>X</p><p> , </p><p>f </p><p>n</p><p> - (</p><p>n</p><p>1</p><p>, </p><p>n</p><p>2</p><p>). </p><p>A (a1 , a2) (a1 + n 1 , a2 + n 2), (a1 - n 1,a2 - n 2).</p><p>3. X - . y = f(x) - X, y - x f(x) = 2x - 6 f(x) = x3. f(x) = x2 - , , , 4 ( 4 ) -: 2 - 2, - 1 .</p><p>4. -- X, , . , ,X x, y, z. f, f(x) = y, f(y) = z f(z) = x, - 1, X 6, .</p><p> X, n , - . - 1 n. ( - , , x 1, y 2, z 3.) f ( - X) : - X ( , 1,, n), f. , f -</p><p> , </p><p> 1 2.</p><p> 2</p><p>, f - X, f </p><p>f = 1 2 32 3 1 </p><p>f 1 = 1 2 32 1 3 </p><p>, f 2 = 1 2 33 2 1 </p><p>, f 3 = 1 2 31 3 2 </p><p>,</p><p>f 4 = 1 2 32 3 1 </p><p>, f 5 = 1 2 33 1 2 </p><p>, f 6 = 1 2 31 2 3 </p><p>.</p><p> . :(1) n = 1, : (1, 2) (2, 1) n = 2 : (2, 1, 3), (3, 2, 1), (1, 3, 2,), (2, 3, 1), (3, 1, 2),(1, 2, 3) n = 3.</p><p> , n X, n , 1 2 3 (n 1) n == n!. ( n .) , - n. , - , 1 n, n + 1 , 1 n + 1, n + 1 - : - , , ..., n-. , - 1 n + 1, n + 1 , , 1 n, , n!(n + 1) = (n + 1)!.</p><p>4. </p><p> - , , , - , .</p><p> f g - X. fg, h, - : x X - h(x) = f(g(x)). , - h(x) y = g(x), z = f(y) y f. , , h - X. - .</p><p>1. f f(x) = 2x + 6 - x. g(x) = x3. h = fg , - </p><p>h(x) = f(g(x)) = f(x3) = 2x3 + 6,</p><p> x x3 , f., h = gf </p><p>h(x) = g(f(x)) = g(2x + 6) = (2x + 6)3.</p><p> , - , - fg gf.</p><p>2. - - 2, f1 f2. - X : X = {1, 2, 3}. ,</p></li><li><p> , 5, 1996118</p><p> h = f1 f2 h(1) = f1( f2(1)). 2 , f2(1) = 3, f1(3) = 3, h(1) = 3. -: h(2) = 1 h(3) = 2. , f1 f2 , f5 2, , f1 f2 = f5. , f2 f1 = f4, f1 f2 , , , f4 f5: f4 f5 = f5 f4.</p><p>3. , , , - 2 4 AB (. 1) 90 , - - 270 . - . - : , ..</p><p>5. </p><p> -. -, - - X. - X. , - , , - ( - , , , - , , -). , , - .</p><p>-, -, - -. : ,, f g , fg, - , , g f., - .</p><p> , ( ) . , ( ) f, g, h X (fg)h = f(gh) ( - ). - , , , , , x X f(g(h(x))). </p><p> , fgh, - . </p><p> , - - e. - x : e(x) = x, , e - ( ) : ef = fe = f f, e(f(x)) = f(x) f (e(x)) = f (x) e. 2 e = f6.</p><p> f g : fg = gf = e; g = f - 1. f - 1, f - 1(b) = a, f(a) = b, f f - 1., (f - 1)- 1 = f. , 2</p><p> - f(x) = x3 f(x) = 2x + 6 . (-!)</p><p> , - f (- ), f - 1 -.</p><p>6. </p><p> 2, - - n x1, x2, , xn, n . Sn 1, 2, ..., n, - n! (. 3). Sn - - n . , f 1 - i1,</p><p>2 i2, ..., n in ( ), -</p><p> x1 , x2 , , xn ., x1 - 2x2 + 5x3x4 , </p><p>, x2 - 2x4 + 5x1x3.</p><p> n = 3 - (x1 + x2)(x1 ++ x3)(x2 + x3) ( ), - . d3(x1, x2, x3) = (x1 - x2)(x1 - x3)(x2 - x3)? , 2,</p><p>f 1- 1 = f 1 f 2- 1 = f 2, f 3- 1 = f 3 f 4- 1 = f 5 f 6- 1 = f 6., , ,</p><p>g x( ) = x3 g x( ) = x2--- 3</p><p>f = 1 2 ni1 i2 in </p><p>xi1 xi2 xin</p><p>1 2 3 4</p><p>2 4 1 3 </p><p>x12</p><p>x22</p><p>x32</p><p>+ +</p></li><li><p> .. 119</p><p> -, . (-!) d3(x1, x2, x3) , -. d3(x1, x2, x3) , -. , f4, f5 f6 - 2. , f1, f2, f3 .</p><p> -</p><p> 1 n, n - x1, x2, , xn :</p><p>dn(x1, x2, , xn) = (x1 - x2)(x1 - x3) (x1 - xn)(x2 - x3) (x2 - xn)(x3 - x4) (x3 - xn) (xn - 1 - xn). (1)</p><p> xk - xl - dn(x1, , xn) , k &lt; l.</p><p> f xk - xl , - (1), , - (1)( k &lt; l, ik &gt; il). , (1) - dn(x1, , xn).</p><p> Sn, - dn(x1, , xn), - (1), , .</p><p> -, -- k k + 1 1, , n:</p><p>(2)</p><p>, (2), - xk - xk + 1 xk + 1 - xk = - (xk - xk + 1). (2) , . , k + 1 &lt; l, xk + 1 - xl xk - xl, - k &lt; l, .. (2) - (1), </p><p> - .</p><p>7. </p><p> , , -, -</p><p>f = 1 2 ni1 i2 in </p><p>xik xil</p><p>f = 1 2 k 1 k k 1+ k 2+ n1 2 k 1 k 1+ k k 2+ n </p><p>.</p><p> . , -, - fg , - g, f. , ( ) , g (1), f dn(x1 , , xn) - dn(x1 , , xn ), fg ( - g, f) dn(x1 , , xn ) dn(x1 , , xn). - . - :</p><p> -. , -, ( ) .</p><p> , , - ( -, ) - k l :</p><p>(3)</p><p> , k &lt; l, , k + 1 &lt; l, (3) , , .</p><p> - g, k + 1 l :</p><p> g - f, (2), - g f g. , - , g f g (3). (, k g , f k + 1, g l. k l, (3), . .) , f . g f g , (3) - . </p><p> 1. - .</p><p> - Sn n &gt; 1 - . , f1, , fs Sn, g1, , gt . </p><p>1 2 k k 1+ l 1 l n1 2 l k 1+ l 1 k n </p><p>.</p><p>g = 1 2 k k 1+ k 2+ l 1 l l 1+ n1 2 k l k 2+ l 1 k 1+ l 1+ n </p><p>.</p></li><li><p> , 5, 1996120</p><p> - - h ( , n &gt; 1). - hf1, , hfs ( hfi = hfj h- 1 - fi = fj). 1 s , , g1, , gt. , s # t. , hg1, , hgt, : t # s. , s = t, 3 , s + t = n!. , </p><p> 2. n &gt; 1 Sn - n!/2.</p><p>8. </p><p> , - -. . , 1 15. - (. 3). , - , , (, , , - ). 3, , - 2 (), 11 (), 6 () 15 ().</p><p>, - , -, 3 3.</p><p> -, - 16, (, , ). </p><p> i1 , ,i2 -, ..., i5 . .</p><p>f = 1 2 16i1 i2 i16 </p><p>,</p><p>, 3 -</p><p> 3 -, 3 -, 14 15.</p><p> , -, . , 3 6 ( 16), f 6 16 . , f g, 6 16. ( -, gf .) 2 ( 11 15) - f , - 2 16 (11 16 15 16 ). </p><p> 1 , , - , .</p><p>, 3 3, . 3 3 , .</p><p> , 3 3 16 . , - , 3 3, , , , . , 3 3 . </p><p> 3 3 (, , !). , - (, , ) , :</p><p> .., .. -. .: , 1985. </p><p>* * *</p><p> , -- , - - - . .. , -. 50 , - - .</p><p>f = 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 163 14 6 8 5 2 16 11 13 4 15 10 9 7 1 12 </p><p>,</p><p>. 3.</p><p>3 14 6 8</p><p>11</p><p>13</p><p>5 2 7</p><p>4</p><p>7</p><p>15 10</p><p>1219</p><p>1 2 3 4</p><p>8</p><p>9</p><p>5 6 7</p><p>10</p><p>15</p><p>11 12</p><p>1413</p><p>1 2 3 4</p><p>8</p><p>9</p><p>5 6 7</p><p>10</p><p>14</p><p>11 12</p><p>161513</p></li></ul>