User bruce westbury - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T12:09:58Z http://mathoverflow.net/feeds/user/3992 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/128522/stratifications-and-cohomology-computations/128598#128598 Answer by Bruce Westbury for Stratifications and Cohomology Computations Bruce Westbury 2013-04-24T10:40:38Z 2013-04-24T10:40:38Z <p>There is the Atiyah-Jones conjecture (now theorem). It even has its own (brief) Wiki page</p> <p><a href="http://en.wikipedia.org/wiki/Atiyah%E2%80%93Jones_conjecture" rel="nofollow">http://en.wikipedia.org/wiki/Atiyah%E2%80%93Jones_conjecture</a></p> http://mathoverflow.net/questions/128512/what-are-the-invariant-definitions-of-spinorial-quantities-from-mathematical-phys/128518#128518 Answer by Bruce Westbury for What are the invariant definitions of spinorial quantities from mathematical physics? Bruce Westbury 2013-04-23T18:29:00Z 2013-04-23T18:29:00Z <p>I don't have the book in front of me but I think a good place to start for this question and similar ones is</p> <p>Quantum Fields and Strings: A Course for Mathematicians by Pierre Deligne</p> <p><a href="http://www.amazon.com/Quantum-Fields-Strings-Course-Mathematicians/dp/0821820125" rel="nofollow">http://www.amazon.com/Quantum-Fields-Strings-Course-Mathematicians/dp/0821820125</a></p> http://mathoverflow.net/questions/127576/what-is-data-involved-in-a-mathematical-construction/127588#127588 Answer by Bruce Westbury for What is "Data" involved in a mathematical construction? Bruce Westbury 2013-04-15T05:33:48Z 2013-04-15T05:33:48Z <p>I read it as saying that: if I (the author) were implementing this on a computer I would write a function that took DATA as input and which returns an instance of whatever we are constructing. In this setting DATA is a finite tuple and we specify what type of thing each term is. Sometimes there are conditions on DATA in which case the function would first check that these conditions are satisfied.</p> <p>If you don't want or like to think in terms of programming it is saying that there is a function from the set of possible DATA (possibly satisfying conditions) to the set of things we are interested in.</p> http://mathoverflow.net/questions/126708/representations-of-symn-and-sl-d/126711#126711 Answer by Bruce Westbury for Representations of Sym(n) and SL_d Bruce Westbury 2013-04-06T15:50:50Z 2013-04-06T15:50:50Z <p>This is given in Exercise 7.74 of Enumerative Combinatorics II by Richard Stanley which is a formula for the restrictions in terms of plethysm and inner product.</p> http://mathoverflow.net/questions/125751/semi-planar-partition-monoid-algebra/125762#125762 Answer by Bruce Westbury for Semi-planar partition monoid/algebra Bruce Westbury 2013-03-27T20:54:29Z 2013-03-28T07:53:18Z <p>I hope I am answering the question:</p> <ol> <li><p>Why not trust the diagrams?</p></li> <li><p>It seems to me that the source of your confusion is that planar refers to the whole diagram and not just the top or bottom row. So planar means that all your conditions are required. You should identify $P_k$ with set partitions of {$1,\cdots ,2k$}.</p></li> <li><p>The planar partition algebra is isomorphic to a Temperley-Lieb algebra. The planar partition algebra arises in the Potts model and the Temperley-Lieb algebras in the ice model. [ You can see this using the diagrams! Start with a Temperley-Lieb diagram with $2k$ points on the top and bottom of a rectangle. This is a non-crossing matching of the $4k$ boundary points. Now colour the regions black and white alternately with the two sides of the rectangle coloured white. Shrink the black intervals on the top and bottom edges to be very short. Then this is an element of $P_k$. ]</p></li> <li><p>The partition algebras map surjectively to the centraliser algebras of the tensor powers of the defining permutation representations of the symmetric groups.</p></li> </ol> http://mathoverflow.net/questions/123227/is-there-a-statistical-interpretation-of-greens-theorem-stokes-theorem-or-the/123314#123314 Answer by Bruce Westbury for Is there a statistical interpretation of Green's theorem, Stokes' theorem, or the divergence theorem? Bruce Westbury 2013-03-01T09:52:29Z 2013-03-01T09:52:29Z <p>I was told that a professor in our department puts it as follows: Your job is to determine the number of cars in a car park. One method is to go around the car park counting them. Alternatively if you know the number at one time then you can stand by the entrance/exit and adjust the number every time a car leaves or arrives.</p> http://mathoverflow.net/questions/123213/decomposition-of-mathrmendv-as-s-n-times-s-n-module/123240#123240 Answer by Bruce Westbury for Decomposition of $\mathrm{End}(V)$ as $S_n\times S_n$-module Bruce Westbury 2013-02-28T16:56:17Z 2013-03-01T09:32:35Z <p>You can give a formula for these numbers in terms of plethysm and internal product and inner product.</p> <p>The starting point is Exercise 7.74 of Enumerative Combinatorics II by Richard Stanley which is a formula for the Schur functors of the defining representation of the symmetric group in terms of plethysm and inner product.</p> http://mathoverflow.net/questions/65691/is-there-a-combinatorial-analogue-of-ricci-flow Is there a combinatorial analogue of Ricci flow? Bruce Westbury 2011-05-22T05:16:29Z 2013-02-08T00:57:27Z <p>The question of generalising circle packing to three dimensions was asked in <a href="http://mathoverflow.net/questions/65677/" rel="nofollow">65677</a>. There is a clear consensus that there is no obvious three dimensional version of circle packing.</p> <p>However I have seen a comment that circle packing on surfaces and Ricci flow on surfaces are related. The circle packing here is an extension of circle packing to include intersection angles between the circles with a particular choice for these angles. My initial question is to ask for an explanation of this.</p> <p>My real question should now be apparent. There is an extension of Ricci flow to three dimensions: so is there some version of circle packing in three dimensions which can be interpreted as a combinatorial version of Ricci flow?</p> http://mathoverflow.net/questions/120671/non-zero-monoid/120705#120705 Answer by Bruce Westbury for Non-zero monoid Bruce Westbury 2013-02-03T21:04:32Z 2013-02-03T21:04:32Z <p>You construct a surjective homomorphism to a non-zero monoid.</p> http://mathoverflow.net/questions/120185/what-are-the-symmetric-and-anti-symmetric-representations-of-6-times6-of-su6/120188#120188 Answer by Bruce Westbury for What are the symmetric and anti-symmetric representations of $6\times6$ of $SU(6)$ in $SU(3)\times SU(2)$? Bruce Westbury 2013-01-29T09:47:01Z 2013-01-29T09:47:01Z <p>First we have $3\times 3 = 6+3$ and $2\times 2 = 3+1$. Then combining this we have</p> <p>$(3,2)\times (3,2) = [ (3,1)+(6,3) ] + [ (3,3)+(6,1) ]$</p> http://mathoverflow.net/questions/118664/is-the-quasisymmetric-expansion-of-the-inner-product-of-two-schur-functions-known Is the quasisymmetric expansion of the inner product of two Schur functions known? Bruce Westbury 2013-01-11T19:18:02Z 2013-01-11T19:18:02Z <p>The question is in the title, however; there is a Hopf algebra of quasisymmetric functions which has the Hopf algebra of symmetric functions as a sub - Hopf algebra. The quasisymmetric functions have a basis of fundamental quasisymmetric functions. The quasisymmetric expansion of a symmetric function means writing it as a linear combination of fundamental quasisymmetric functions.</p> <p>The inner product of two homogeneous symmetric functions is only nonzero if they have the same degree. Let $\chi$ and $\chi'$ be two characters of the symmetric group $S_r$. Let $\mathrm{ch}$ be the characteristic map. The inner product $\ast$ is characterised by $\mathrm{ch}(\chi)\ast\mathrm{ch}(\chi')=\mathrm{ch}(\chi\chi')$.</p> <p>I am interested in the inner product of two Schur functions. This is a symmetric function and so is a linear combination of Schur functions. It is a known open problem to give a combinatorial interpretation of these coefficients.</p> <p>I am asking a variation of this question, namely, is anything known about the quasisymmetric expansion of the inner product of two Schur functions?</p> http://mathoverflow.net/questions/118536/delignes-1996-note-on-exceptional-lie-groups/118547#118547 Answer by Bruce Westbury for Deligne's 1996 note on exceptional Lie groups Bruce Westbury 2013-01-10T16:12:01Z 2013-01-10T16:12:01Z <p>My understanding is that the series corresponding to the first three rows of the magic square do not exist but the existence of the series for the last row, the original exceptional series, is still open. This follows from Dylan Thurston's (unpublished) computer calculation and from Pierre Vogel's (unpublished) papers.</p> http://mathoverflow.net/questions/117943/littlewood-richardson-rule-and-seminormal-basis-of-specht-modules/117974#117974 Answer by Bruce Westbury for Littlewood Richardson rule and seminormal basis of Specht modules Bruce Westbury 2013-01-03T16:00:58Z 2013-01-03T16:00:58Z <p>These references solve the analogous problem for the general linear groups:</p> <pre><code>MR2166314 (2006h:20062) Howe, Roger E. ; Tan, Eng-Chye ; Willenbring, Jeb F. A basis for the GLn tensor product algebra. Adv. Math. 196 (2005), no. 2, 531--564. MR2888167 Howe, Roger ; Lee, Soo Teck . Why should the Littlewood-Richardson rule be true? Bull. Amer. Math. Soc. (N.S.) 49 (2012), no. 2, 187--236. MR0955587 (89j:20046) Tokuyama, Takeshi . Determinantal method and the Littlewood-Richardson rule. J. Algebra 117 (1988), no. 1, 1--18. </code></pre> <p>with some work, using Schur-Weyl duality, you should be able to solve your problem.</p> http://mathoverflow.net/questions/117847/a-lie-subgroup-of-so4n/117853#117853 Answer by Bruce Westbury for A lie Subgroup of SO(4n) Bruce Westbury 2013-01-02T09:23:08Z 2013-01-02T09:23:08Z <p>For all $n,m\ge 0$ we have a homomorphism $Sp(n) \times Sp(m)/(\pm \mathrm{Id})\rightarrow SO(4nm)$. This is just the tensor product of the two defining representations.</p> http://mathoverflow.net/questions/117724/is-there-formula-name-and-proof-for-this-theorem/117732#117732 Answer by Bruce Westbury for Is there formula name and proof for this theorem ? Bruce Westbury 2012-12-31T14:14:48Z 2012-12-31T14:14:48Z <p>The formula can be found in:</p> <p>MR2036721 (2005b:14068) Lando, Sergei K. ; Zvonkin, Alexander K. </p> <p>Graphs on surfaces and their applications.</p> <p>With an appendix by Don B. Zagier.</p> <p>Encyclopaedia of Mathematical Sciences, 141. Low-Dimensional Topology, II.</p> <p>Springer-Verlag, Berlin, 2004. xvi+455 pp. ISBN: 3-540-00203-0 </p> http://mathoverflow.net/questions/116069/exact-arithmetic-for-real-algebraic-numbers Exact arithmetic for real algebraic numbers Bruce Westbury 2012-12-11T10:56:41Z 2012-12-12T13:51:39Z <p>There was a reply to a question (that I can't find) which mentioned SARAG (Some Algorithms in Real Algebraic Geometry) see <a href="http://perso.univ-rennes1.fr/marie-francoise.roy/bpr-ed2-posted2.html" rel="nofollow">http://perso.univ-rennes1.fr/marie-francoise.roy/bpr-ed2-posted2.html</a>. This is a package for Maxima.</p> <p>I looked into this and came across the Thom encoding of real algebraic numbers. If I have understood correctly, you are given a square-free polynomial with, say integer coefficients, and a real root. Then evaluate the derivatives of the polynomial at the root to get a sequence of signs. The polynomial together with the sign sequence is an encoding of the root.</p> <p>My question is whether you can implement the operations of an ordered field on these encodings? The operation I not clear on is the ordering. A specific question is: given an encoding as above and a polynomial; evaluate the polynomial at the real number. Is there an algorithm for deciding if this is positive, negative, (or zero)?</p> <p>I assume it is possible to evaluate this numerically using interval arithmetic and that if you did this with sufficient accuracy you would eventually arrive at a solution. However this seems clumsy and I am asking for something more effective.</p> <p>As an illustration of why this could be useful. There was a question on the word problem in a Coxeter group (which again I could not find). One solution to the word problem is to look at the image of a fixed vector in the geometric representation. Then you can determine the descent set of a word by looking at which entries are negative. This is fine theoretically but not if you work with fixed precision real numbers as the entries will rapidly become very small in absolute value. A positive solution to my question would make this an effective algorithm.</p> http://mathoverflow.net/questions/111754/finding-generators-of-matrix-subgroups/115604#115604 Answer by Bruce Westbury for Finding generators of matrix subgroups Bruce Westbury 2012-12-06T11:41:34Z 2012-12-06T11:41:34Z <p>For the specific example you mention there are these references:</p> <p>MR0513734 (80k:20036) Assion, Joachim . Einige endliche Faktorgruppen der Zopfgruppen. (German) Math. Z. 163 (1978), no. 3, 291--302.</p> <p>MR1177345 (93m:20053) Wajnryb, Bronislaw . A braidlike presentation of Sp(n,p). Israel J. Math. 76 (1991), no. 3, 265--288.</p> <p>These give presentations for prime fields but I believe you can deduce presentations for the integers.</p> http://mathoverflow.net/questions/113911/is-fg-wr-s-n-a-cellular-algebra/113923#113923 Answer by Bruce Westbury for Is $F[G\wr S_n]$ a cellular algebra ? Bruce Westbury 2012-11-20T10:05:37Z 2012-11-20T10:05:37Z <p>I don't believe that $F[G]$ is cellular in general. My reason (and this is not a proof) is that cellular algebras are typically quasi-hereditary (which implies finite cohomological dimension) and group algebras are not close to quasi-hereditary.</p> http://mathoverflow.net/questions/96069/status-of-pl-topology/111714#111714 Answer by Bruce Westbury for Status of PL topology Bruce Westbury 2012-11-07T09:27:59Z 2012-11-07T09:27:59Z <p>On a smooth manifold we have Ricci flow. What is the analogue for a PL manifold?</p> http://mathoverflow.net/questions/109470/directed-graphs-and-compact-lie-groups/109477#109477 Answer by Bruce Westbury for Directed graphs and Compact Lie Groups Bruce Westbury 2012-10-12T18:15:20Z 2012-10-12T18:15:20Z <p>This is a very broad question and there is a lot I could say on the topic. I suggest as a starting point you read the book:</p> <p>Cvitanović, Predrag . Group theory. Birdtracks, Lie's, and exceptional groups. Princeton University Press, Princeton, NJ, 2008. </p> http://mathoverflow.net/questions/108890/tensor-powers-of-the-standard-representation/108894#108894 Answer by Bruce Westbury for Tensor powers of the standard representation Bruce Westbury 2012-10-05T07:19:42Z 2012-10-05T07:19:42Z <p>You want the "partition algebras". Some references to get you started are:</p> <p>MR1317365 (97b:82023) Jones, V. F. R. The Potts model and the symmetric group. Subfactors (Kyuzeso, 1993), 259--267, World Sci. Publ., River Edge, NJ, 1994. </p> <p>MR1399030 (98g:05152) Martin, Paul . The structure of the partition algebras. J. Algebra 183 (1996), no. 2, 319--358.</p> <p>MR2143201 (2006g:05228) Halverson, Tom ; Ram, Arun . Partition algebras. European J. Combin. 26 (2005), no. 6, 869--921.</p> <p>The partition algebras are the endomorphism algebras of the tensor powers of, $V$, the natural representation of $S_n$. As has been mentioned in the comments this decomposes as the sum of the trivial representation and the representation you are interested in.</p> <p>You can recover information about the representation you are interested in from the partition algebras. For example, instead of looking at all set partitions, you only consider set partitions with no singleton.</p> http://mathoverflow.net/questions/107340/invariants-of-group-action-sl-n-acts-simultaneously-on-m-symmetric-matrices/107363#107363 Answer by Bruce Westbury for Invariants of group action: SL_n acts simultaneously on m symmetric matrices Bruce Westbury 2012-09-17T07:33:09Z 2012-09-17T10:28:35Z <p>If you are willing to replace $SL$ by $GL$ (so without the determinant) then this is a special case of a much more general result.</p> <p>Apply the theory in the following paper to the quiver with one vertex and $m$ edges.</p> <p>MR0958897 (90e:16048) Le Bruyn, Lieven; Procesi, Claudio</p> <p>Semisimple representations of quivers.</p> <p>Trans. Amer. Math. Soc. 317 (1990), no. 2, 585–598.</p> <p>The special case you are considering was studied before this. The reference has already been given by Agol.</p> http://mathoverflow.net/questions/106804/a-generalization-of-catalan-numbers/106807#106807 Answer by Bruce Westbury for A generalization of Catalan numbers Bruce Westbury 2012-09-10T11:48:26Z 2012-09-10T11:48:26Z <p>Maybe you are looking for the Fuss-Catalan numbers?</p> <p>These are given by $\frac{1}{(s-1)n+1}\binom{sn}{n}$.</p> <p>A starting point for the combinatorial significance of these numbers is:</p> <p><a href="http://arxiv.org/abs/math/9811086" rel="nofollow">http://arxiv.org/abs/math/9811086</a></p> http://mathoverflow.net/questions/106046/phase-choice-for-6j-symbols/106056#106056 Answer by Bruce Westbury for Phase choice for 6j symbols Bruce Westbury 2012-08-31T17:26:39Z 2012-09-02T11:54:29Z <p>There are two real forms of $SL(2,C)$, the compact real form $SU(2)$ and the split real form $SL(2,R)$. Mathematicians prefer the split real form and physicists the compact real form. The two cases have different signs. There is also a choice of sign conventions involved even once you have decided which real form you want.</p> <p>Edit: There is another choice of sign. The two dimensional representation has an invariant symplectic form. Alternatively you can take this to be a super vector space with no even part in which case it has an invariant symmetric inner product. You need the latter convention for diagrams with unoriented edges.</p> http://mathoverflow.net/questions/105318/fictive-irreps-of-the-enveloping-general-lie-algebra/105326#105326 Answer by Bruce Westbury for "Fictive" irreps of the enveloping general Lie algebra Bruce Westbury 2012-08-23T15:07:11Z 2012-08-23T15:07:11Z <p>The (quantum) dimensions are all given in Vogel's papers (all but one unpublished). There may be some errors in my paper you refer to as I confused Vogel's parameters with the values of the quadratic Casimir.</p> <p>I don't follow your "silly part". The representation $V$ is not defined for $E_8$ and I don't see how you get a proposed quantum dimension.</p> http://mathoverflow.net/questions/105151/mini-fusion-categories-via-6j-symbols/105162#105162 Answer by Bruce Westbury for "Mini" fusion categories via 6j symbols Bruce Westbury 2012-08-21T15:00:15Z 2012-08-21T15:00:15Z <p>Fusion categories are discrete (like finite groups) and you never have a "free parameter". This is an observation by Ocneanu.</p> http://mathoverflow.net/questions/103020/which-polynomials-are-fricke-polynomials/103038#103038 Answer by Bruce Westbury for Which polynomials are Fricke polynomials ? Bruce Westbury 2012-07-24T18:24:16Z 2012-07-24T20:07:52Z <p><S>You have not shown that your map $t$ is well-defined. I suspect you are defining the quotient of $SL_2\times SL_2$ by simultaneous conjugation. This was studied by Procesi.</S></p> <p>Edit: A quick search found the paper <a href="http://www.mathematics.jhu.edu/brown/Documents/FrickeCharAutos.pdf" rel="nofollow">http://www.mathematics.jhu.edu/brown/Documents/FrickeCharAutos.pdf</a> which confirms Andreas comment. It also gives the action of $Out(F_2)$ in Example 6.1. This raises the question: does $Out(F_2)$ act transitively on this set of polynomials?</p> <p>In particular they give the reference:</p> <p>`Fricke, R., and Klein, F., Vorlesungen uber die Theorie der automorphem Functionen, Vol. 1, pp. 365-370. Leipzig: B.G. Teubner 1897. Reprint: New York Juhnson Reprint Corporation (Academic Press) 1965</p> <p>This can be found at <a href="http://archive.org/details/vorlesungenber01fricuoft" rel="nofollow">http://archive.org/details/vorlesungenber01fricuoft</a></p> http://mathoverflow.net/questions/102432/uniform-sampling-hemisphere-and-project-in-a-specific-direction/102433#102433 Answer by Bruce Westbury for Uniform sampling hemisphere and project in a specific direction Bruce Westbury 2012-07-17T09:12:45Z 2012-07-17T09:12:45Z <p>This is fully discussed in</p> <p><a href="http://mathoverflow.net/questions/24688" rel="nofollow">http://mathoverflow.net/questions/24688</a></p> http://mathoverflow.net/questions/102207/form-of-elements-of-a-lie-algebra/102211#102211 Answer by Bruce Westbury for Form of elements of a Lie algebra Bruce Westbury 2012-07-14T08:20:16Z 2012-07-14T13:46:00Z <p>The key words are "Lyndon words". A reference is Free Lie Algebras by C. Reutenauer.</p> http://mathoverflow.net/questions/102154/finite-subgroups-of-spin9/102167#102167 Answer by Bruce Westbury for Finite subgroups of Spin(9) Bruce Westbury 2012-07-13T18:58:35Z 2012-07-13T18:58:35Z <p>Although there exists an algorithm which will list all finite subgroups of $Spin(9)$ I suspect it is not effective. For background and references see</p> <p><a href="http://mathoverflow.net/questions/17072" rel="nofollow">http://mathoverflow.net/questions/17072</a></p> <p>I don't know if imposing the condition that the group acts freely on $S^{15}$ improves the situation. </p> http://mathoverflow.net/questions/130574/why-do-i-get-estimated-error-1-ind-when-doing-bicgstab-linear-solver-using-il Comment by Bruce Westbury Bruce Westbury 2013-05-14T13:22:06Z 2013-05-14T13:22:06Z Maybe your matrix is singular? http://mathoverflow.net/questions/129857/relating-two-characterizations-of-mathfrak-sl-n-2-among-simple-lie-algeb Comment by Bruce Westbury Bruce Westbury 2013-05-06T17:15:11Z 2013-05-06T17:15:11Z @Jim I have edited, hopefully, to what OP intended (as Terry Tao suggested). http://mathoverflow.net/questions/128873/tits-kantor-koecher-construction-for-jordan-algebra-of-symmetric-bilinear-form Comment by Bruce Westbury Bruce Westbury 2013-04-27T06:58:11Z 2013-04-27T06:58:11Z Is $R$ the Clifford algebra of the form you start with? http://mathoverflow.net/questions/128538/man-standing-on-circumference-of-a-circle Comment by Bruce Westbury Bruce Westbury 2013-04-23T20:58:01Z 2013-04-23T20:58:01Z Please read the FAQ before posting. This is not a site for puzzles. http://mathoverflow.net/questions/127540/does-induction-for-a-functor-algebra-imply-it-is-initial Comment by Bruce Westbury Bruce Westbury 2013-04-14T18:03:35Z 2013-04-14T18:03:35Z @Andrej Comment deleted http://mathoverflow.net/questions/126270/can-all-knot-polynomials-derived-from-skein-relations-be-categorified Comment by Bruce Westbury Bruce Westbury 2013-04-02T15:35:19Z 2013-04-02T15:35:19Z Can the categorified Jones polynomial be derived from skein relations? http://mathoverflow.net/questions/125943/degree-of-a-commutator-in-a-hyperalgebra-or-enveloping-algebra Comment by Bruce Westbury Bruce Westbury 2013-03-29T21:23:47Z 2013-03-29T21:23:47Z Isn't $X=Y$ a counterexample to both your proposals? http://mathoverflow.net/questions/123213/decomposition-of-mathrmendv-as-s-n-times-s-n-module/123240#123240 Comment by Bruce Westbury Bruce Westbury 2013-03-01T09:44:51Z 2013-03-01T09:44:51Z It takes a while to get used to plethysm and it is notoriously difficult to compute. http://mathoverflow.net/questions/120875/ring-with-three-binary-operations Comment by Bruce Westbury Bruce Westbury 2013-02-05T17:07:47Z 2013-02-05T17:07:47Z @darij You left out &quot;second plethysm&quot;, usually called &quot;inner plethysm&quot; http://mathoverflow.net/questions/120612/trichotomies-in-mathematics/120752#120752 Comment by Bruce Westbury Bruce Westbury 2013-02-05T06:07:20Z 2013-02-05T06:07:20Z Do you mean prime knots? http://mathoverflow.net/questions/120184/how-to-write-down-explictly-the-isomorphism-of-two-finite-dimensional-representat Comment by Bruce Westbury Bruce Westbury 2013-01-29T12:41:49Z 2013-01-29T12:41:49Z Look up &quot;Koszul complex&quot; http://mathoverflow.net/questions/120084/what-kinds-of-operations-are-well-defined-when-working-with-sets-classes-conglo Comment by Bruce Westbury Bruce Westbury 2013-01-28T07:34:41Z 2013-01-28T07:34:41Z The axioms are what you say they are. http://mathoverflow.net/questions/120030/representation-theoretic-foundations-for-facts-about-binary-forms-and-their-duals Comment by Bruce Westbury Bruce Westbury 2013-01-27T16:35:27Z 2013-01-27T16:35:27Z See <a href="http://mathoverflow.net/questions/45664/" rel="nofollow">mathoverflow.net/questions/45664</a> http://mathoverflow.net/questions/120030/representation-theoretic-foundations-for-facts-about-binary-forms-and-their-duals Comment by Bruce Westbury Bruce Westbury 2013-01-27T16:34:05Z 2013-01-27T16:34:05Z See <a href="http://mathoverflow.net/questions/15336/" rel="nofollow">mathoverflow.net/questions/15336</a> http://mathoverflow.net/questions/120030/representation-theoretic-foundations-for-facts-about-binary-forms-and-their-duals Comment by Bruce Westbury Bruce Westbury 2013-01-27T16:33:31Z 2013-01-27T16:33:31Z See <a href="http://mathoverflow.net/questions/34452/" rel="nofollow">mathoverflow.net/questions/34452</a>