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While ", FontVariations->{"CompatibilityType"->0}], StyleBox["Mathematica,", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}], StyleBox[" which aims to be a general purpose numeric-symbolic program, \ can never be as efficient a tool for research level symbolic computation \ algebra as more specialized programs, the dramatic increase in the speed of \ personal computers has made such use entirely feasible. Moreover, the \ elegance and ease of the ", FontVariations->{"CompatibilityType"->0}], StyleBox["Mathematica", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}], StyleBox[" programming language make it often easier to implement \ mathematical tools far removed form the interests of the majority of ", FontVariations->{"CompatibilityType"->0}], StyleBox["Mathematica", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}], StyleBox[" users and even ", FontVariations->{"CompatibilityType"->0}], StyleBox["Mathematica", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}], StyleBox[" designers. \n\n We choose as an example a rather arcane subject: \ the computation of the action of the Steenrod operations in cohomology of the \ classifying space of the exceptional Lie group ", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ \(TraditionalForm\`F\_4\)]], ". The computation of the cohomology of the classifying spaces of \ exceptional Lie groups was began by Toda ([T1],[T2]) is still incomplete and \ an active area of research. The computation we perform has been done by \ hand by Vavpetic [V] and required a certain amount of manipulation and \ mathematical cleverness. Here we shall use nothing but the \"brute strength\" \ of ", StyleBox["Mathematica", FontSlant->"Italic"], ". In order to perform our computation we have to implement the Steenrod \ squaring operations in mod 2 cohomology. The simplicity and elegance of the \ mathematica programming language make such an implementation almost trivial. \ The computation we perform here is quite complex and would be impossible to \ perform \"by hand\" directly (although of course human intelligence is still \ far more powerful mathematical tool than the most powerful computers armed \ with the best symbolic algebra programs). " }], "Text"], Cell["", "Text"], Cell["\<\ Summary of basic facts from Algebraic Topology \ \>", "Subtitle"], Cell[TextData[{ "Lie groups, which generalize groups of invertible matrices and operators \ are the basic objects of study in several areas of mathematics, as well as \ being of fundamental importance in physics. A Lie group is simultaneously a \ topological object (a smooth manifold) and an algebraic one (group) and these \ two structures are compatible in the sense that the usual algebraic \ operations (group multiplication, taking the inverse) are given by smooth \ maps. The simplest Lie groups are of course finite groups which have trivial \ topological structure. Their natural generalization are compact Lie groups. \ These have been completely classified (via their associated Lie algebras) \ into several families (each with infinitely many members) and several so \ called \"exceptional\" Lie groups, denoted by ", Cell[BoxData[ \(TraditionalForm\`G\_2, \ F\_4\)]], ", ", Cell[BoxData[ \(TraditionalForm\`E\_6\)]], ",", Cell[BoxData[ \(TraditionalForm\`E\_7\)]], ",", Cell[BoxData[ \(TraditionalForm\`E\_8\)]], ". Now, let ", Cell[BoxData[ \(TraditionalForm\`G\)]], " be a compact Lie group. Associated with each such group there is a \ topological space ", Cell[BoxData[ \(TraditionalForm\`B\ G\)]], ". The characterization of such spaces is an important problem in algebraic \ topology. An step in that direction is the computation of their cohomology \ rings. A famous theorem of Armand Borel reduces the question to invariant \ theory." }], "Text"], Cell[TextData[{ StyleBox["Theorem", FontWeight->"Bold"], ".(Borel) Let ", Cell[BoxData[ \(TraditionalForm\`G\)]], " be a compact connected Lie group with maximal torus ", Cell[BoxData[ \(TraditionalForm\`T\)]], " and the Weyl group ", Cell[BoxData[ \(TraditionalForm\`\(\(W\_G\)\(.\)\)\)]], " If ", Cell[BoxData[ \(TraditionalForm\`F\)]], " is a field of characteristic relatively prime to the order of ", Cell[BoxData[ \(TraditionalForm\`W\_G\)]], " then the projection map ", Cell[BoxData[ \(TraditionalForm\`B\ T\ \[LongRightArrow]\ B\ G\)]], "induces an isomorphism ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(\(\(H\^*\)(B\ G, \ F)\)\(\ \)\(\[TildeFullEqual]\)\), "TraditionalForm"], \(\(\(H\^*\)(B\ T, \ F)\)\^W\_G\)}], TraditionalForm]]], ".\n" }], "Theorem"], Cell[TextData[{ "This theorem makes it possible to compute cohomology groups of compact Lie \ groups with rational coefficients and with coefficients in finite fields of \ characteristic relatively prime to the Weyl group. However, the case of \ field coefficients with characteristic dividing the order of the Weyl group \ is far from simple and is an active area of research. It is also an area in \ which computer aided algebraic computations are very useful. Most of the \ computations involve computing the cohomology of the manifold ", Cell[BoxData[ \(TraditionalForm\`G\)]], " and passing to the cohomology of ", Cell[BoxData[ \(TraditionalForm\`B\ G\)]], " using the fibration ", Cell[BoxData[ \(TraditionalForm\`G\ \[LongRightArrow]\ E\ G\ \[LongRightArrow] B\ G\)]], " and the associated Moore-Eilenberg spectral sequence. However, there is \ another approach which makes use of invariant theory. In his doctoral thesis \ Vavpetic [V] computed the cohomology of the exceptional Lie group ", Cell[BoxData[ \(TraditionalForm\`F\_4\)]], " with coefficients in ", Cell[BoxData[ \(TraditionalForm\`\[DoubleStruckCapitalF]\_2\)]], "using this approach. The basic idea of the proof is to consider a maximal \ elementary abelian 2-subgroup of ", Cell[BoxData[ \(TraditionalForm\`\(\(F\_4\)\(.\)\(\ \)\)\)]], "One can show that up to conjugation there is just one such group ", Cell[BoxData[ \(TraditionalForm\`E\_32 = \((\[DoubleStruckCapitalZ]/2)\)\^5\)]], ". By the Weyl group of ", Cell[BoxData[ \(TraditionalForm\`E\_32\)]], "we mean the quotient ", Cell[BoxData[ \(TraditionalForm\`\(W\_\(F\_4\)\)(E\_32) = \(\(N\_\(F\_4\)\)(E\_32)\)/ E\_32\)]], "of the normalizer of ", Cell[BoxData[ \(TraditionalForm\`E\_32\)]], " in ", Cell[BoxData[ \(TraditionalForm\`F\_4\)]], ". The Weyl group ", Cell[BoxData[ \(TraditionalForm\`\(W\_\(F\_4\)\)(E\_32)\)]], " can be viewed as a subgroup of ", Cell[BoxData[ FormBox[ RowBox[{\(GL\_5\), "(", \(\[DoubleStruckCapitalF]\_2\), StyleBox[")", FontFamily->"Geneva"]}], TraditionalForm]]], " acting on the vector space ", Cell[BoxData[ \(TraditionalForm\`\[DoubleStruckCapitalF]\_2\^5\)]], " by linear automorphism. This induces and action on the cohomology ", Cell[BoxData[ \(TraditionalForm\`\(H\^*\)( B\ \((\[DoubleStruckCapitalZ]/2)\)\^5, \ \[DoubleStruckCapitalF]\_2)\)]], " which is the polynomial algebra ", Cell[BoxData[ FormBox[ RowBox[{\(\[DoubleStruckCapitalF]\_2\), StyleBox["[", FontFamily->"Geneva"], RowBox[{ SubscriptBox[ StyleBox["t", FontFamily->"Geneva"], "1"], StyleBox[",", FontFamily->"Geneva", FontSlant->"Italic"], StyleBox[" ", FontFamily->"Geneva", FontSlant->"Italic"], RowBox[{ StyleBox[".", FontFamily->"Geneva", FontSlant->"Italic"], StyleBox[" ", FontFamily->"Geneva", FontSlant->"Italic"], StyleBox[".", FontFamily->"Geneva", FontSlant->"Italic"], StyleBox[" ", FontFamily->"Geneva", FontSlant->"Italic"], StyleBox[".", FontFamily->"Geneva", FontSlant->"Italic"], StyleBox[" ", FontFamily->"Geneva", FontSlant->"Italic"], \(t\_5\)}]}], StyleBox["]", FontFamily->"Geneva"]}], TraditionalForm]]], ". Vavpetic has shown that the subalgebra of invariants (fixed points) ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{\(\[DoubleStruckCapitalF]\_2\), StyleBox["[", FontFamily->"Geneva"], RowBox[{ StyleBox[ RowBox[{ StyleBox["t", FontFamily->"Geneva", FontSlant->"Italic"], StyleBox["1", FontFamily->"Geneva", FontSize->8]}]], StyleBox[",", FontFamily->"Geneva", FontSlant->"Italic"], StyleBox[" ", FontFamily->"Geneva", FontSlant->"Italic"], RowBox[{ StyleBox[".", FontFamily->"Geneva", FontSlant->"Italic"], StyleBox[" ", FontFamily->"Geneva", FontSlant->"Italic"], StyleBox[".", FontFamily->"Geneva", FontSlant->"Italic"], StyleBox[" ", FontFamily->"Geneva", FontSlant->"Italic"], StyleBox[".", FontFamily->"Geneva", FontSlant->"Italic"], StyleBox[" ", FontFamily->"Geneva", FontSlant->"Italic"], StyleBox[ RowBox[{ StyleBox["t", FontFamily->"Geneva", FontSlant->"Italic"], StyleBox["5", FontFamily->"Geneva", FontSize->8, FontSlant->"Italic"]}]]}]}], StyleBox["]", FontFamily->"Geneva"]}], \(\(W\_\(F\_4\)\)(E\_32)\)], TraditionalForm]]], "is isomorphic to the cohomology of ", Cell[BoxData[ \(TraditionalForm\`\(\(B\)\(\ \)\(F\_4\)\(\ \)\)\)]], ". He also explicitly constructed the invariants that generate the \ cohomology ring.\nIn addition to having the structure of a graded ring the \ cohomology groups ", Cell[BoxData[ \(TraditionalForm\`\(H\^*\)(X, \[DoubleStruckCapitalF]\_2)\)]], "of a topological space ", Cell[BoxData[ \(TraditionalForm\`X\)]], " ", Cell[BoxData[ \(TraditionalForm\`\ \)]], "admit the action of certain natural operations, known as the Steenrod \ squares [St]. The action of these operations is a powerful invariant \ (meaning that topologically equivalent spaces not only have isomorphic \ cohomology rings but also the Steenrod operations act in the same way). \ Multiplicative properties of the Steenrod operations (the so called \"Cartan \ formula\") imply that one only needs to compute their action on the \ generators of the cohomology ring. This is what we propose to do. First \ however, we need to describe these generators and implement the Steenrod \ operations in ", StyleBox["Mathematica", FontSlant->"Italic"], ". \n" }], "Text"], Cell["Notation", "Subtitle"], Cell[TextData[{ "\nWe would like to use the usual superscript notation ", Cell[BoxData[ \(TraditionalForm\`Sq\^i\)]], " for the ", Cell[BoxData[ \(TraditionalForm\`i\)]], "-th Steenrod square operation. However, in ", StyleBox["Mathematica", FontSlant->"Italic"], " superscripts are already reserved for denoting powers, and treating the \ ", Cell[BoxData[ \(TraditionalForm\`i\)]], "-th Steenrod operation as an ", Cell[BoxData[ \(TraditionalForm\`i\)]], "-th power would cause problems. We avoid this by means of the Notation \ package. This will allow us to use the superscript notation for inputting the \ ", Cell[BoxData[ \(TraditionalForm\`i\)]], "-th Steenrod square while at the same time using the notation ", Cell[BoxData[ \(TraditionalForm\`Sq[i]\)]], " when making definitions, which avoids the above mentioned problems. In \ addition we format the first Steenrod operation ", Cell[BoxData[ \(TraditionalForm\`Sq\^1\)]], " as ", Cell[BoxData[ \(TraditionalForm\`\[Beta]\)]], ", which is the usual thing to do (", Cell[BoxData[ \(TraditionalForm\`\(\(\[Beta]\)\(\ \)\)\)]], "is also known as \"the Bockstein\")." }], "Text"], Cell[BoxData[ \(<< "\"\)], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"Notation", "[", RowBox[{ TagBox[\(Sq\^i_\), NotationBoxTag, TagStyle->"NotationTemplateStyle"], " ", "\[DoubleLongLeftRightArrow]", " ", TagBox[\(Sq[i_]\), NotationBoxTag, TagStyle->"NotationTemplateStyle"]}], "]"}], ";"}]], "Input"], Cell[BoxData[ \(\(\(Format[Sq[1]] := \[Beta];\)\(\n\) \)\)], "Input"], Cell["\<\ Dickson Invariants\ \>", "Subtitle"], Cell[TextData[{ "The algebra of invariants of ", Cell[BoxData[ FormBox[ RowBox[{\(GL\_n\), "(", \(\[DoubleStruckCapitalF]\_2\), StyleBox[")", FontFamily->"Geneva"]}], TraditionalForm]]], " was calculated by Dickson, who showed that " }], "Text"], Cell[BoxData[ \(\[DoubleStruckCapitalF]\_2[t\_1, ... , t\_n]\^GL\_n[\ \[DoubleStruckCapitalF]\_2] = \[DoubleStruckCapitalF]\_2[c\_\(n, 0\) ... , c\_\(n, n - 1\)]\)], "Equation"], Cell[TextData[{ "The generators ", Cell[BoxData[ \(TraditionalForm\`\(\(c\_\(n, i\)\)\(\ \)\)\)]], "are known as Dickson invariants. They are given by the following formula:" }], "Text"], Cell[BoxData[ \(\(\(\(f\_n\) \((X)\) = \(\[Product]\+\(\[Nu] = \[Sum]\+\(i = 1\)\%n\( k\ \_i\) t\_i\)\((X - \[Nu])\) = \(X\^\(q\^n\) + \[Sum]\+\(i = 0\)\%\(n - \ 1\)\(\((\(-1\))\)\^\(n - i\)\) \(c\_\(n, i\)\) X\^\(q\^i\) = 0\)\)\)\(,\)\)\)], "Equation"], Cell[TextData[{ "where ", Cell[BoxData[ \(TraditionalForm\`k\_i\)]], "are either 1 or 0. " }], "Text"], Cell["We implement Dickson invariants as follows.", "Text"], Cell[BoxData[ \(c[n_, i_, t_ : t, k_ : 0] := Block[{ls = Distribute[Table[{0, 1}, {n}], List], x}, PolynomialMod[ Coefficient[ Times @@ \((x - \((Array[Subscript[t, k + #1] &, {n}] . #1 &)\) /@ ls)\), x\^\(2\^i\)], 2]]\)], "Input"], Cell["\<\ The third argument is the letter we use to denote the variables in \ our invariant. By always take that to be t but other letters are possible. \ The fourth argument determines the initial value of the index of this \ variable. For example\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(c[3, 2]\)], "Input"], Cell[BoxData[ \(t\_1\%4 + t\_1\%2\ t\_2\%2 + t\_2\%4 + t\_1\%2\ t\_2\ t\_3 + t\_1\ t\_2\%2\ t\_3 + t\_1\%2\ t\_3\%2 + t\_1\ t\_2\ t\_3\%2 + t\_2\%2\ t\_3\%2 + t\_3\%4\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(c[3, 2, t, 1]\)], "Input"], Cell[BoxData[ \(t\_2\%4 + t\_2\%2\ t\_3\%2 + t\_3\%4 + t\_2\%2\ t\_3\ t\_4 + t\_2\ t\_3\%2\ t\_4 + t\_2\%2\ t\_4\%2 + t\_2\ t\_3\ t\_4\%2 + t\_3\%2\ t\_4\%2 + t\_4\%4\)], "Output"] }, Open ]], Cell["\<\ Dickson invariants tend to be very complicated. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(c[4, 2]\)], "Input"], Cell[BoxData[ \(t\_1\%8\ t\_2\%4 + t\_1\%4\ t\_2\%8 + t\_1\%8\ t\_2\%2\ t\_3\%2 + t\_1\%2\ t\_2\%8\ t\_3\%2 + t\_1\%8\ t\_3\%4 + t\_1\%4\ t\_2\%4\ t\_3\%4 + t\_2\%8\ t\_3\%4 + t\_1\%4\ t\_3\%8 + t\_1\%2\ t\_2\%2\ t\_3\%8 + t\_2\%4\ t\_3\%8 + t\_1\%8\ t\_2\%2\ t\_3\ t\_4 + t\_1\%2\ t\_2\%8\ t\_3\ t\_4 + t\_1\%8\ t\_2\ t\_3\%2\ t\_4 + t\_1\ t\_2\%8\ t\_3\%2\ t\_4 + t\_1\%2\ t\_2\ t\_3\%8\ t\_4 + t\_1\ t\_2\%2\ t\_3\%8\ t\_4 + t\_1\%8\ t\_2\%2\ t\_4\%2 + t\_1\%2\ t\_2\%8\ t\_4\%2 + t\_1\%8\ t\_2\ t\_3\ t\_4\%2 + t\_1\ t\_2\%8\ t\_3\ t\_4\%2 + t\_1\%8\ t\_3\%2\ t\_4\%2 + t\_1\%4\ t\_2\%4\ t\_3\%2\ t\_4\%2 + t\_2\%8\ t\_3\%2\ t\_4\%2 + t\_1\%4\ t\_2\%2\ t\_3\%4\ t\_4\%2 + t\_1\%2\ t\_2\%4\ t\_3\%4\ t\_4\%2 + t\_1\%2\ t\_3\%8\ t\_4\%2 + t\_1\ t\_2\ t\_3\%8\ t\_4\%2 + t\_2\%2\ t\_3\%8\ t\_4\%2 + t\_1\%8\ t\_4\%4 + t\_1\%4\ t\_2\%4\ t\_4\%4 + t\_2\%8\ t\_4\%4 + t\_1\%4\ t\_2\%2\ t\_3\%2\ t\_4\%4 + t\_1\%2\ t\_2\%4\ t\_3\%2\ t\_4\%4 + t\_1\%4\ t\_3\%4\ t\_4\%4 + t\_1\%2\ t\_2\%2\ t\_3\%4\ t\_4\%4 + t\_2\%4\ t\_3\%4\ t\_4\%4 + t\_3\%8\ t\_4\%4 + t\_1\%4\ t\_4\%8 + t\_1\%2\ t\_2\%2\ t\_4\%8 + t\_2\%4\ t\_4\%8 + t\_1\%2\ t\_2\ t\_3\ t\_4\%8 + t\_1\ t\_2\%2\ t\_3\ t\_4\%8 + t\_1\%2\ t\_3\%2\ t\_4\%8 + t\_1\ t\_2\ t\_3\%2\ t\_4\%8 + t\_2\%2\ t\_3\%2\ t\_4\%8 + t\_3\%4\ t\_4\%8\)], "Output"] }, Open ]], Cell["\<\ We shall be working in graded rings, so we assign all the \ indeterminates the degree 1.\ \>", "Text"], Cell[BoxData[ \(\(dim[Subscript[t, i_]] = 1;\)\)], "Input"], Cell["", "Text"], Cell["\<\ Working mod 2 \ \>", "Subtitle"], Cell[TextData[{ "We shall need to use the Steenrod operations in cohomology. These exists \ whenever the coefficient are taken in a field of finite characterisitc, but \ we shall only consider the case of coefficients in ", Cell[BoxData[ \(TraditionalForm\`F\_2\)]], ". We would like to work mod 2 but also to be able to switch to \ characteristic 0 when convenient. First, we make a list of functions which \ accept the option Modulus:\n" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(functions = ToExpression /@ Select[DeleteCases[ Names["\"], _?\((StringTake[#1, 1] == "\<$\>" &)\)], \(! FreeQ[Options[Symbol[#1]], Modulus]\) &]\)], "Input"], Cell[BoxData[ \({Apart, ApartSquareFree, Cancel, Coefficient, CoefficientList, Collect, Decompose, Denominator, Det, Expand, ExpandAll, ExpandDenominator, ExpandNumerator, Exponent, Factor, FactorList, FactorSquareFree, FactorSquareFreeList, FactorTerms, FactorTermsList, GroebnerBasis, Inverse, LinearSolve, LUBackSubstitution, LUDecomposition, NullSpace, Numerator, PolynomialGCD, PolynomialLCM, PolynomialMod, PolynomialQ, PolynomialReduce, Resultant, Roots, RowReduce, Together, TrigReduce, Variables}\)], "Output"] }, Open ]], Cell["\<\ The next function will automatically set the modulus to p for all \ the function that accept this option.\ \>", "Text"], Cell[BoxData[ \(FunctionsMod[p_] := Scan[SetOptions[#1, Modulus \[Rule] p] &, functions]\)], "Input"], Cell["\<\ We can now switch all the functions in the above list to working \ modulo 2 and back to characteristic 0 (or any other).\ \>", "Text"], Cell["Here is how it works", "Text"], Cell[BoxData[ \(FunctionsMod[2]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Expand[\((x + 1)\)\^4]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`x\^4 + 1\)], "Output"] }, Open ]], Cell[BoxData[ \(FunctionsMod[0]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Expand[\((x + 1)\)\^4]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`x\^4 + 4\ x\^3 + 6\ x\^2 + 4\ x + 1\)], "Output"] }, Open ]], Cell["In what follows we work mod 2. ", "Text"], Cell[BoxData[ \(FunctionsMod[2]\)], "Input"], Cell["\<\ The Steenrod Squares\ \>", "Subtitle"], Cell[TextData[{ "Next we implement the Steenrod operations . This implementation contains \ several superfluous rules, whose presence however seems to improve \ performance. The implementation is obtained basically just by copying the \ properties of the Steenrod operations from any text on algebraic topology and \ writing them down using ", StyleBox["Mathematica", FontSlant->"Italic"], " syntax. The naturalness and easy of this construction seem to me \ unmatched by other symbolic algebra programs. The last two rules are the \ famous Cartan and Adem formulas. (The latter one is not needed for the \ computations performed below, but we include it for completeness)." }], "Text"], Cell[BoxData[{ \(\(c[n_, m_] := Mod[\(n!\)\/\(\(\((n - m)\)!\)\ \(m!\)\), 2];\)\), "\n", \(\(\(Sq[l_List]\)[x_] := Fold[\(Sq[#2]\)[#1] &, x, l];\)\), "\n", \(\(dim[\(Sq[i_]\)[x_]] := i + dim[x];\)\), "\n", \(\(Sq[0] = Identity;\)\), "\n", \(\(\(Sq[1]\)[\(Sq[1]\)[x_]] := 0;\)\), "\n", \(\(\(Sq[1]\)[\(Sq[i_?EvenQ]\)[x_]] := \(Sq[i + 1]\)[x];\)\), "\n", \(\(\(Sq[1]\)[\(Sq[i_?OddQ]\)[x_]] := 0;\)\), "\n", \(\(\(Sq[i_]\)[0] = 0;\)\), "\n", \(\(\(Sq[i_]\)[k_Integer\ x_] := Mod[k, 2]\ \(Sq[i]\)[x];\)\), "\n", \(\(Sq /: \(Sq[i_]\)[x_ + y_] := PolynomialMod[\(Sq[i]\)[x] + \(Sq[i]\)[y], 2];\)\), "\n", \(\(dim[k_?IntegerQ] := 0;\)\), "\n", \(\(dim[x_\ y_] := dim[x] + dim[y];\)\), "\n", \(\(dim[x_\^n_] := n\ dim[x];\)\), "\n", \(\(dim[\(Sq[i_]\)[x_]] := i + dim[x];\)\), "\n", \(\(\(Sq[i_]\)[0] := 0;\)\), "\n", \(\(\(Sq[i_]\)[a_] /; dim[a] < i := 0;\)\), "\n", \(\(\(Sq[i_]\)[a_] /; dim[a] == i := a\^2;\)\), "\n", \(\(\(Sq[i_]\)[x_\^k_Integer?Positive] /; dim[x] == 1 := c[k, i]\ x\^\(i + k\);\)\), "\n", \(\(\(Sq[k_]\)[x_\ y_] := PolynomialMod[\[Sum]\+\(i = 0\)\%k\( Sq[i]\)[x]\ \(Sq[k - i]\)[y], 2];\)\), "\[IndentingNewLine]", \(\(\(Sq[a_]\)[\(Sq[b_]\)[x_]] /; 0 < a < 2\ b := PolynomialMod[\[Sum]\+\(j = 0\)\%\(Floor[a\/2]\)c[b - 1 - j, a - 2\ j]\ \(Sq[a + b - j]\)[\(Sq[j]\)[x]], 2];\)\)}], "Input"], Cell["\<\ Computation of the action of Steenrod Squares on the invariants.\ \ \>", "Subtitle"], Cell[TextData[{ "\nWe now apply this to compute the action of the Steenrod operations on \ the cohomology of the exceptional Lie group ", Cell[BoxData[ \(TraditionalForm\`F\_4\)]], ". " }], "Text"], Cell[TextData[{ "However, it is first of all necessary to construct the invariants in ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{\(\[DoubleStruckCapitalF]\_2\), StyleBox["[", FontFamily->"Geneva"], RowBox[{ StyleBox[ RowBox[{ StyleBox["t", FontFamily->"Geneva", FontSlant->"Italic"], StyleBox["1", FontFamily->"Geneva", FontSize->8]}]], StyleBox[",", FontFamily->"Geneva", FontSlant->"Italic"], StyleBox[" ", FontFamily->"Geneva", FontSlant->"Italic"], RowBox[{ StyleBox[".", FontFamily->"Geneva", FontSlant->"Italic"], StyleBox[" ", FontFamily->"Geneva", FontSlant->"Italic"], StyleBox[".", FontFamily->"Geneva", FontSlant->"Italic"], StyleBox[" ", FontFamily->"Geneva", FontSlant->"Italic"], StyleBox[".", FontFamily->"Geneva", FontSlant->"Italic"], StyleBox[" ", FontFamily->"Geneva", FontSlant->"Italic"], SubscriptBox[ StyleBox["t", FontFamily->"Geneva", FontSlant->"Italic"], "5"]}]}], StyleBox["]", FontFamily->"Geneva"]}], \(\(W\_\(F\_4\)\)(E\_32)\)], TraditionalForm]]], " that generate the cohomology ring. Vavpetic constructs such invariants \ out of Dickson invariants in ", Cell[BoxData[ FormBox[ SuperscriptBox[ FormBox[ RowBox[{\(\[DoubleStruckCapitalF]\_2\), StyleBox["[", FontFamily->"Geneva"], RowBox[{ SubscriptBox[ StyleBox["t", FontFamily->"Geneva"], "3"], StyleBox[",", FontFamily->"Geneva", FontSlant->"Italic"], SubscriptBox[ StyleBox["t", FontFamily->"Geneva", FontSlant->"Italic"], "4"], StyleBox[" ", FontFamily->"Geneva", FontSlant->"Italic"], StyleBox[",", FontFamily->"Geneva", FontSlant->"Italic"], StyleBox[" ", FontFamily->"Geneva", FontSlant->"Italic"], SubscriptBox[ StyleBox["t", FontFamily->"Geneva", FontSlant->"Italic"], "5"]}], StyleBox["]", FontFamily->"Geneva"]}], "TraditionalForm"], \(GL(3, \[DoubleStruckCapitalF]\_3)\)], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ SuperscriptBox[ FormBox[ RowBox[{\(\[DoubleStruckCapitalF]\_2\), StyleBox["[", FontFamily->"Geneva"], RowBox[{ SubscriptBox[ StyleBox["t", FontFamily->"Geneva"], "1"], StyleBox[",", FontFamily->"Geneva", FontSlant->"Italic"], SubscriptBox[ StyleBox["t", FontFamily->"Geneva", FontSlant->"Italic"], "2"]}], StyleBox[" ", FontFamily->"Geneva", FontSlant->"Italic"], StyleBox["]", FontFamily->"Geneva"]}], "TraditionalForm"], \(GL(2, \[DoubleStruckCapitalF]\_3)\)], TraditionalForm]]], ". The general nature of this construction is explained in [K]. In fact \ the invariance of the elements constructed by Vavpetic can be also verified \ by a direct ", StyleBox["Mathematica", FontSlant->"Italic"], " computation, but as this is very time consuming here we only implement \ them in ", StyleBox["Mathematica", FontSlant->"Italic"], " without any further comment." }], "Text"], Cell[TextData[{ "First of all, we define the following three generators of degree 4, 6 and \ 7 , in which are simply Dickson invariants, that is, invariants of the action \ of ", Cell[BoxData[ FormBox[ RowBox[{\(GL\_3\), "(", \(\[DoubleStruckCapitalF]\_2\), StyleBox[")", FontFamily->"Geneva"]}], TraditionalForm]]], " on ", Cell[BoxData[ FormBox[ RowBox[{\(\[DoubleStruckCapitalF]\_2\), StyleBox["[", FontFamily->"Geneva"], RowBox[{ SubscriptBox[ StyleBox["t", FontFamily->"Geneva"], "3"], StyleBox[",", FontFamily->"Geneva", FontSlant->"Italic"], SubscriptBox[ StyleBox["t", FontFamily->"Geneva", FontSlant->"Italic"], "4"], StyleBox[" ", FontFamily->"Geneva", FontSlant->"Italic"], StyleBox[",", FontFamily->"Geneva", FontSlant->"Italic"], StyleBox[" ", FontFamily->"Geneva", FontSlant->"Italic"], SubscriptBox[ StyleBox["t", FontFamily->"Geneva", FontSlant->"Italic"], "5"]}], StyleBox["]", FontFamily->"Geneva"]}], TraditionalForm]]], ". By thinking of ", Cell[BoxData[ FormBox[ RowBox[{\(GL\_3\), "(", \(\[DoubleStruckCapitalF]\_2\), StyleBox[")", FontFamily->"Geneva"]}], TraditionalForm]]], " as a subgroup of ", Cell[BoxData[ FormBox[ RowBox[{\(GL\_5\), "(", \(\[DoubleStruckCapitalF]\_2\), StyleBox[")", FontFamily->"Geneva"]}], TraditionalForm]]], ", which acts trivially on ", Cell[BoxData[ \(TraditionalForm\`t\_1\)]], "and ", Cell[BoxData[ \(TraditionalForm\`t\_2\)]], "we obtain three invariant elements of degree 4, 6 and 7. " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(y\_4 = c[3, 2, t, 2]\)], "Input"], Cell[BoxData[ \(t\_3\%4 + t\_3\%2\ t\_4\%2 + t\_4\%4 + t\_3\%2\ t\_4\ t\_5 + t\_3\ t\_4\%2\ t\_5 + t\_3\%2\ t\_5\%2 + t\_3\ t\_4\ t\_5\%2 + t\_4\%2\ t\_5\%2 + t\_5\%4\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(y\_6 = c[3, 1, t, 2]\)], "Input"], Cell[BoxData[ \(t\_3\%4\ t\_4\%2 + t\_3\%2\ t\_4\%4 + t\_3\%4\ t\_4\ t\_5 + t\_3\ t\_4\%4\ t\_5 + t\_3\%4\ t\_5\%2 + t\_3\%2\ t\_4\%2\ t\_5\%2 + t\_4\%4\ t\_5\%2 + t\_3\%2\ t\_5\%4 + t\_3\ t\_4\ t\_5\%4 + t\_4\%2\ t\_5\%4\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(y\_7 = c[3, 0, t, 2]\)], "Input"], Cell[BoxData[ \(t\_3\%4\ t\_4\%2\ t\_5 + t\_3\%2\ t\_4\%4\ t\_5 + t\_3\%4\ t\_4\ t\_5\%2 + t\_3\ t\_4\%4\ t\_5\%2 + t\_3\%2\ t\_4\ t\_5\%4 + t\_3\ t\_4\%2\ t\_5\%4\)], "Output"] }, Open ]], Cell[TextData[{ "\nTwo more invariants, of degree 16 and 24 and involving both ", Cell[BoxData[ \(TraditionalForm\`t\_1\)]], "and ", Cell[BoxData[ \(TraditionalForm\`t\_2\)]], "are defined as follows:" }], "Text"], Cell[BoxData[ \(a\_i_ := t\_i\%8 + y\_4\ t\_i\%4 + y\_6\ t\_i\%2 + y\_7\ t\_i\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ FormBox[\(y\_16\), "TraditionalForm"], "=", RowBox[{ SuperscriptBox[ FormBox[\(a\_1\), "TraditionalForm"], "2"], "+", RowBox[{ FormBox[\(a\_2\), "TraditionalForm"], " ", FormBox[\(a\_1\), "TraditionalForm"]}], "+", SuperscriptBox[ FormBox[\(a\_2\), "TraditionalForm"], "2"]}]}], ";"}], TraditionalForm]], "Input"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(y\_24\), "=", RowBox[{ FormBox[\(a\_1\), "TraditionalForm"], " ", FormBox[\(a\_2\), "TraditionalForm"], " ", RowBox[{"(", RowBox[{ FormBox[\(a\_1\), "TraditionalForm"], "+", FormBox[\(a\_2\), "TraditionalForm"]}], ")"}]}]}], ";"}], TraditionalForm]], "Input"], Cell["\<\ We do not print out these as they are represented by rather long expressions, \ even with mod 2 coefficients, indeed we have:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Length[PolynomialMod[Expand[y\_24], 2]]\)], "Input"], Cell[BoxData[ \(614\)], "Output"] }, Open ]], Cell["\<\ Of course we can now simply apply the Steenrod operations to our \ invariants directly and even get some useful results, such as\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(Sq\^2\)[y\_7]\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell["However", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(Sq\^4\)[y\_7]\)], "Input"], Cell[BoxData[ \(t\_3\%8\ t\_4\%2\ t\_5 + t\_3\%2\ t\_4\%8\ t\_5 + t\_3\%8\ t\_4\ t\_5\%2 + t\_3\ t\_4\%8\ t\_5\%2 + t\_3\%2\ t\_4\ t\_5\%8 + t\_3\ t\_4\%2\ t\_5\%8\)], "Output"] }, Open ]], Cell[TextData[{ "is not what we want, since we would like to express each invariant \ generator in terms of the other ones. Of course this is just the job for \ GroebnerBasis and PolynomialReduce [CLS]. The problem is that we can't get \ our output in terms of the ", Cell[BoxData[ \(TraditionalForm\`y\_i\)]], " since ", StyleBox["Mathematica", FontSlant->"Italic"], " will automatically express them in terms of the ", Cell[BoxData[ \(TraditionalForm\`t\_i\)]], ". To cope with this we introduce new variables, ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[ RowBox[{ FormBox[\(Y\_4\), "TraditionalForm"], ",", \(Y\_6\)}], "TraditionalForm"], ",", \(Y\_7\)}], TraditionalForm]]], ", ", Cell[BoxData[ \(TraditionalForm\`Y\_16\)]], ", ", Cell[BoxData[ \(TraditionalForm\`Y\_24\)]], "which we think of as new set of names for ", Cell[BoxData[ \(y\_4\)]], ", ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[ RowBox[{ FormBox[\(y\_6\), "TraditionalForm"], ",", \(y\_7\)}], "TraditionalForm"], ",", \(y\_16\)}], TraditionalForm]]], "and ", Cell[BoxData[ \(TraditionalForm\`y\_24\)]], " (If we wished to make it look as if we were using the same symbols we \ could define ", Cell[BoxData[ \(TraditionalForm\`\(\(Y\_4\)\(=\)\(HoldForm[y\_4]\)\(\ \)\)\)]], "etc.) We can then express the action of the Steenrod squares on the little \ y's in terms of the large Y's." }], "Text"], Cell[BoxData[ \(ideal1 = {Y\_4 - c[3, 2, t, 2], Y\_6 - c[3, 1, t, 2], Y\_7 - c[3, 0, t, 2]}; \)], "Input"], Cell[BoxData[ \(ideal2 = Join[ideal1, {Y\_16 - y\_16, Y\_24 - y\_24}]; \)], "Input"], Cell[BoxData[ \(vars1 = Join[Table[t[i], {i, 3, 5}], {Y\_4, Y\_6, Y\_7}]; \)], "Input"], Cell[BoxData[ \(vars2 = Join[Table[t[i], {i, 1, 5}], {Y\_4, Y\_6, Y\_7, Y\_16, Y\_24}]; \)], "Input"], Cell[BoxData[ \(groebner1 = GroebnerBasis[ideal1, vars1]; \)], "Input"], Cell[BoxData[ \(groebner2 = GroebnerBasis[ideal2, vars2]; \)], "Input"], Cell["\<\ Now we can compute the actions of the Steenrod operations on the \ invariants (of course we only need to compute the action of Steenrod squares \ in dimensions lees than the dimension of the invariant, the remaining squares \ act trivially by definition). \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ FormBox[ RowBox[{"Table", "[", RowBox[{ RowBox[{ RowBox[{"PolynomialReduce", "[", RowBox[{ RowBox[{ FormBox[\(Sq\^\(2\^i\)\), "TraditionalForm"], "(", \(y\_4\), ")"}], ",", "groebner1", ",", "vars1"}], "]"}], "\[LeftDoubleBracket]", "2", "\[RightDoubleBracket]"}], ",", \({i, 0, 2}\)}], "]"}], TraditionalForm]], "Input"], Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{"0", ",", FormBox[\(Y\_6\), "TraditionalForm"], ",", SuperscriptBox[ FormBox[\(Y\_4\), "TraditionalForm"], "2"]}], "}"}], TraditionalForm]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ FormBox[ RowBox[{"Table", "[", RowBox[{ RowBox[{ RowBox[{"PolynomialReduce", "[", RowBox[{ RowBox[{ FormBox[\(Sq\^\(2\^i\)\), "TraditionalForm"], "(", \(y\_6\), ")"}], ",", "groebner1", ",", "vars1"}], "]"}], "\[LeftDoubleBracket]", "2", "\[RightDoubleBracket]"}], ",", \({i, 0, 2}\)}], "]"}], TraditionalForm]], "Input"], Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{ FormBox[\(Y\_7\), "TraditionalForm"], ",", "0", ",", RowBox[{ FormBox[\(Y\_4\), "TraditionalForm"], " ", FormBox[\(Y\_6\), "TraditionalForm"]}]}], "}"}], TraditionalForm]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ FormBox[ RowBox[{"Table", "[", RowBox[{ RowBox[{ RowBox[{"PolynomialReduce", "[", RowBox[{ RowBox[{ FormBox[\(Sq\^\(2\^i\)\), "TraditionalForm"], "(", \(y\_7\), ")"}], ",", "groebner1", ",", "vars1"}], "]"}], "\[LeftDoubleBracket]", "2", "\[RightDoubleBracket]"}], ",", \({i, 0, 2}\)}], "]"}], TraditionalForm]], "Input"], Cell[BoxData[ \(TraditionalForm\`{0, 0, Y\_4\ Y\_7}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ FormBox[ RowBox[{"Table", "[", RowBox[{ RowBox[{ RowBox[{"PolynomialReduce", "[", RowBox[{ RowBox[{ FormBox[\(Sq\^\(2\^i\)\), "TraditionalForm"], "(", \(y\_16\), ")"}], ",", "groebner2", ",", "vars2"}], "]"}], "\[LeftDoubleBracket]", "2", "\[RightDoubleBracket]"}], ",", \({i, 0, 4}\)}], "]"}], TraditionalForm]], "Input"], Cell[BoxData[ \(TraditionalForm\`{0, 0, 0, Y\_16\ Y\_4\%2 + Y\_24, Y\_16\%2}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ FormBox[ RowBox[{"Table", "[", RowBox[{ RowBox[{ RowBox[{"PolynomialReduce", "[", RowBox[{ RowBox[{ FormBox[\(Sq\^\(2\^i\)\), "TraditionalForm"], "(", \(y\_24\), ")"}], ",", "groebner2", ",", "vars2"}], "]"}], "\[LeftDoubleBracket]", "2", "\[RightDoubleBracket]"}], ",", \({i, 0, 4}\)}], "]"}], TraditionalForm]], "Input"], Cell[BoxData[ \(TraditionalForm\`{0, 0, Y\_4\ Y\_24, Y\_4\%2\ Y\_24, Y\_4\ Y\_24\ Y\_6\%2 + Y\_16\ Y\_24}\)], "Output"] }, Open ]], Cell["\<\ These results agree with those obtained by Vavpetic.\ \>", "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["References", "Section"], Cell[TextData[{ StyleBox["[CLS]\[ThickSpace]\[MediumSpace]D. Cox, J. Little, D. O'Shea, \ Using Algebraic Geometry, Springer-Verlag, 1998.", "Item1"], "\n[K]\[ThickSpace]\[MediumSpace]M. ", StyleBox["Kameko, Rings of Invariants derived from Dickson Invariants \ (pre-print)", "Reference"], "\n[S]\[ThickSpace]\[MediumSpace]L. ", StyleBox["Smith, Polynomial Invariants of Finite groups, A K Peters Ltd, \ 1995.", "Reference"], "\n[St]\[ThickSpace]\[MediumSpace]N. E. ", StyleBox["Steenrod, Cohomology Operations, written and revised by D.B.A. \ Epstein, Princeton University Press, 1962", "ReferenceTwo"], "\n[T1] \[ThickSpace]H. \[MediumSpace]", StyleBox["Toda, Cohomology of classifying spaces of exceptional Lie groups, \ Manifolds, pp. 265-271, Tokyo 1973", "Reference"], "\n[T2]\[ThickSpace]\[MediumSpace]", StyleBox["H. Toda, Cohomology mod 3 of the classifying space ", "Reference"], Cell[BoxData[ \(TraditionalForm\`BF\_4\)]], " of the exceptional Lie group ", Cell[BoxData[ \(TraditionalForm\`F\_4\)]], ", J. Math. Kyoto Univ. 13 (1972), 97-115\n[V]\[ThickSpace]\[MediumSpace]A. \ ", StyleBox["Vavpetic. Homotopy Characterization of classifying spaces of \ compact Lie groups. University of Lubljana doctoral thesis, 2000.", "ReferenceTwo"], "\n[W]\[ThickSpace]\[MediumSpace]C. ", StyleBox["Wilkerson, A primer on the Dickson Invariants, a corrected \ version of the paper published in Contemp. 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