User alexander moll - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T21:27:02Z http://mathoverflow.net/feeds/user/6862 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/124923/evaluating-an-infinite-product-of-q-exponentials Evaluating an infinite product of q-exponentials Alexander Moll 2013-03-19T03:11:42Z 2013-04-02T18:04:40Z <p>For the $q$-exponential $$e_q(u) = \sum_{n=0}^{\infty} \frac{u^n}{[n]_q!}$$ with $[k]_q=\frac{1-q^k}{1-q}$ and $[n]_q! = [n]_q [n-1]_q \cdots [1]_q$, we don't have the property $e_q(u) e_q(v) = e_q (u+v)$. I learned this the hard way when trying to compute the infinite product $$\prod_{i=0}^{\infty} e_q (x q^i)$$ where $x$ is an indeterminate.</p> <ul> <li>Does anyone see a way to get a closed form expression for this product?</li> </ul> <p>Any help would be greatly appreciated!</p> http://mathoverflow.net/questions/123032/two-appearances-of-the-jacobi-triple-product-identity Two appearances of the Jacobi Triple Product identity Alexander Moll 2013-02-26T20:34:16Z 2013-02-26T21:38:05Z <p>So far, I've seen two ways of deriving the Jacobi Triple Product (JTP) formula $$\sum_{m \in \mathbb{Z}} (-u)^m q^{\frac{m(m-1)}{2}}= \prod_{j=0}^{\infty} (1-uq^{j})(1-q^{j+1})(1-u^{-1} q^{j+1})$$ using infinite-dimensional representation theory. On the one hand, it is the Weyl-Kac denominator formula for $\widehat{\mathfrak{sl}_2}$. On the other hand, it is the character $$\text{Tr}(q^{L_0} u^{\alpha_0})$$ computed in two ways using the boson-fermion correspondence. Here $$\alpha_0 = \sum_{i \in \mathbb{Z}'} : \psi_i \psi_i^* :$$ is the zero mode of the Heisenberg algebra which computes the charge of the Dirac sea of free fermions, and $$L_0 = \sum_{i \in \mathbb{Z}'} i : \psi_i \psi_i^* :$$ is the energy operator for a particular representation of the Virasoro algebra on this semi-infinite wedge space.</p> <ul> <li>Is there a structural relationship between these two appearances of JTP?</li> </ul> http://mathoverflow.net/questions/92840/bochners-theorem-and-total-positivity Bochner's Theorem and Total Positivity Alexander Moll 2012-04-01T19:17:01Z 2012-08-15T23:22:01Z <p>Bochner's Theorem for LCA groups applied to the case of $G = U(1)$ and $G^{\vee} = \mathbb{Z}$ tells us that through the Fourier transform, probability measures on the circle are in bijection with infinite positive semidefinite matrices with $1$s along the main diagonal. In the case of finite $N \times N$ matrices, we know from convex analysis that these are <em>correlation matrices</em>. Indeed, this corresponds to the case $G = \mathbb{Z} / N \mathbb{Z} \hookrightarrow U(1)$ of $N$th roots of unity with its dual group $\mathbb{Z} \twoheadrightarrow \mathbb{Z} / N \mathbb{Z} = G^{\vee}$.</p> <p>Concretely, every <em>principal</em> minor of a positive semidefinite matrix has non-negative determinant. If a matrix satisfies the stronger condition that <em>every</em> minor has non-negative determinant, we call it a <em>totally positive</em> matrix. </p> <ul> <li>Is there some nice condition on a positive semidefinite matrix which guarantees it is totally positive?</li> <li>Which probability measures on the circle correspond to infinite totally positive matrices with $1$s on the main diagonal?</li> </ul> http://mathoverflow.net/questions/77014/recovering-the-alexander-polynomial-from-ocneanus-homflypt Recovering the Alexander Polynomial from Ocneanu's HOMFLYPT Alexander Moll 2011-10-03T06:14:50Z 2012-05-23T01:09:57Z <p>Let $H_q(n)$ denote the Hecke algebra associated to the symmetric group $S(n)$: this is the $\mathbb{Z}[q^{\pm 1/2}]$ algebra generated by $T_1, \ldots, T_{n-1}$ satisfying the braid relations along with $T_i^2=(q-1)T_i + q$. Ocneanu's trace $\tau_z (T_{\omega_{\mu}}) = z^{l(\omega_{\mu})}$ defined on fundamental elements is the unique normalized trace on our Hecke algebra that can be jiggled to yield invariants of oriented links: this gives the two-parameter HOMFLYPT polynomials $P_L(q,z)$. These polynomials satisfy the skein relation</p> <p>$$\Big ( \frac{z}{z-q+1} \Big )^{1/2} P_{L_+} (q,z) - \Big ( \frac{z}{z-q+1} \Big )^{-1/2} P_{L_-}(q,z) = (q^{1/2} - q^{-1/2}) P_{L_0} (q,z).$$</p> <p>which motivates the common change of variables $x = \sqrt{\frac{z}{z-q+1}}$, $y = q^{1/2} - q^{-1/2}$. Note that the target ring of this trace has to be at least $\mathbb{Z}[q^{\pm 1/2}, z^{\pm 1/2}, (z-q+1)^{ \pm 1/2}]$ if we want to write down the associated invariant $P_L(q,z)$.</p> <p>Many people take the HOMFLYPT polynomials to be those obtained after the specialization $z=q^{N}/[N]$, which seems to be equivalent to $x=\sqrt{\frac{z}{z-q+1}} = q^{N/2}$. Setting $N=2$ recovers the Jones polynomial, and setting $N=0$ is supposed to recover the Alexander polynomial. </p> <ul> <li>How am I supposed to correctly obtain the Alexander polynomial in terms of the original $q$ and $z$?</li> </ul> <p>It seems that $x=q^{0/2}=1$ gives the correct specialization. Still, doesn't $\sqrt{\frac{z}{z-q+1}}=1$ force $q=1$, leaving $z$ free? How does the right-hand side of the skein relation survive then?</p> http://mathoverflow.net/questions/92736/two-curious-asymptotic-results-for-dimensions-of-type-a-objects Two curious asymptotic results for dimensions of type A objects Alexander Moll 2012-03-31T06:35:33Z 2012-04-03T14:42:48Z <p>Let $V_{\lambda}$ and $W_{\lambda}$ be the irreducible representations of $S(n)$ and $\mathfrak{su}(N,\mathbb{C})$ associated to the partition $\lambda \in \mathbb{Y}$ of size $| \lambda |=n$ and length $l(\lambda) \leq N$. The following limit $$\frac{\dim V_{\lambda}}{n!} = \lim_{N \rightarrow \infty} \frac{\dim W_{\lambda}}{N^{n}}$$ follows immediately from the well known hook (content) formulas $$\dim V_{\lambda} = \prod_{\square \in \lambda} \frac{n!}{h(\square)} \ \ \ \ \dim W_{\lambda} = \prod_{\square \in \lambda} \frac{N + c(\square)}{h(\square)}$$ which can be found in Macdonald's book. Notice that $n! = \dim_{\mathbb{C}} \mathbb{C}[S(n)]$ and $N^n = \dim_{\mathbb{C}} (\mathbb{C}^N)^{\otimes n}$, so what we're seeing is that as $N \rightarrow \infty$, the relative multiplicity of $V_{\lambda}$ in Schur-Weyl duality approaches the relative multiplicity of $V_{\lambda}$ in the regular representation. </p> <p>Does anyone have a good feeling for why this is true?</p> <p>Also, let us not forget the Peter-Weyl theorem! If for a compact group $G$ we write $G^{\vee}$ for its set of finite dimensional irreducible representations over $\mathbb{C}$, we have $$L^2(SU(N)) = \widehat{\bigoplus_{\lambda \in SU(N)^{\vee}}} W_{\lambda} \boxtimes W_{\lambda}$$ $$(\mathbb{C}^N)^{\otimes n}=\bigoplus_{\lambda \in SU(N)^{\vee} \cap S(n)^{\vee}} V_{\lambda} \boxtimes W_{\lambda}$$ $$\mathbb{C}[S(n)] = \bigoplus_{\lambda \in S(n)^{\vee}} V_{\lambda} \boxtimes V_{\lambda}$$</p> <p>The limit we discussed above relating the second to the third line here actually also happens when we pass from the first to the second line: the relative multiplicity'' of $W_{\lambda}$ in its regular representation approaches the relative multiplicity of $W_{\lambda}$ in Schur-Weyl duality.</p> <p>Can anyone give me some intuition for what's going on here + why I might expect such a result?</p> http://mathoverflow.net/questions/88645/whitehead-products-and-a-realization-problem-for-graded-lie-algebras Whitehead products and a realization problem for graded Lie algebras Alexander Moll 2012-02-16T16:15:09Z 2012-02-16T16:15:09Z <p>Many $\mathbb{Z}$-graded Lie algebras $\mathfrak{g}$ over $\mathbb{C}$ we would like to study are <em>non-degenerate</em> in the sense that </p> <ol> <li>$\dim_{\mathbb{C}} \mathfrak{g}_n &lt; \infty \ \forall n \in \mathbb{Z}$</li> <li>$\mathfrak{g}_0$ is abelian</li> <li>For generic $\lambda \in \mathfrak{g}_0^*$, $\forall n \in \mathbb{N}$ the bilinear form $\mathfrak{g_n} \otimes \mathfrak{g}_{-n} \rightarrow \mathbb{C}$ given by $$a \otimes b \mapsto \lambda ([a,b])$$ is non-degenerate.</li> </ol> <p>This includes simple Lie algebras, Loop algebras, affine Kac-Moody algebras, the Heisenberg algebra, the Witt algebra, and the Virasoro algebra.</p> <p>For any of these examples, can we find some topological space $X$ such that $\pi_{*+1}(X) \cong \mathfrak{g}_{\geq 0}$ under the Whitehead bracket? </p> http://mathoverflow.net/questions/81035/coincidences-amongst-classifying-spaces-and-eilenberg-mac-lane-spaces Coincidences amongst classifying spaces and Eilenberg Mac-Lane spaces Alexander Moll 2011-11-16T04:17:50Z 2011-12-05T12:37:37Z <p>Given that $$\mathbb{R}P^{\infty} = B O(1) = K(\widehat{O(1)}, 1)$$ $$\mathbb{C} P^{\infty} = B U(1) = K( \widehat{U(1)}, 2)$$ is there any way to make sense of $$\mathbb{H}P^{\infty} = B Sp(1)$$ in a similar manner using the representation theory of the non-abelian group $Sp(1) \cong Spin(3) \cong SU(2)$? </p> http://mathoverflow.net/questions/81024/taylor-expansion-of-a-q-analog-of-the-negative-binomial-distribution Taylor expansion of a q-analog of the negative binomial distribution Alexander Moll 2011-11-16T00:40:17Z 2011-11-16T01:42:37Z <p>Given $A,B \in \mathbb{Z}_+$ and $0 &lt; t, q&lt; 1$, I'd like to compute the coefficients $c_n(q,A,B)$ in the expansion of the product $$\prod_{i=0}^{A-1} \prod_{j=0}^{B-1} \frac{1}{1-t q^{i+j}} = \sum_{n=0}^{\infty} c_n t^n.$$ As $q \rightarrow 1$, this returns the well known formula $$\frac{1}{(1-t)^{AB}} = \sum_{n=0}^{\infty} \binom{AB+n-1}{n} t^n$$ which has a quick enumerative proof.</p> <p>So far I've determined that the highest power of $q$ in $c_n(q,A,B)$ is $(A-1)(B-1)n$ less than the highest power of $q$ in the $q$-binomial coefficient $\binom{AB+n-1}{n}_q$. Can anyone see what these $c_n$ count in the expansion of the product? Any help would be greatly appreciated!</p> http://mathoverflow.net/questions/77884/does-there-exist-a-complex-lie-group-g-such-that Does there exist a complex Lie group G such that ... Alexander Moll 2011-10-12T01:50:50Z 2011-10-12T16:47:58Z <p>... every Riemann surface of genus $1$ appears as a complex one-parameter subgroup of $G$?</p> http://mathoverflow.net/questions/76597/analytic-characterization-of-parallel-transport-of-fundamental-groups Analytic Characterization of Parallel Transport of Fundamental Groups Alexander Moll 2011-09-28T03:43:35Z 2011-09-29T02:23:29Z <p>(Note that I've edited the main body of the question to make it clear for other readers.)</p> <p>Fix a principal $G$-bundle $\rho: P \rightarrow X$ and fix a point $p \in P_x$ in the fiber above $x \in X$. If $\rho$ is equipped with a flat connection $\omega$, we get a surjective homomorphism $\pi_1(X,x) \rightarrow \text{Hol}_p(\omega)$. </p> <ul> <li>When is the map $\pi_1(X, x) \rightarrow \text{Hol}_p(\omega) \hookrightarrow G \rightarrow \text{Inn}(G)$ surjective?</li> </ul> <p>The motivation for this question comes from the fact that such a composition is surjective in the case of parallel transport'' of fundamental groups in $X$ along curves changing basepoints.</p> http://mathoverflow.net/questions/75998/two-ways-of-generalizing-factorials-via-symmetric-groups Two ways of generalizing factorials via symmetric groups Alexander Moll 2011-09-20T21:28:26Z 2011-09-21T04:52:41Z <p>By the Bruhat decomposition of $GL(n, \mathbb{F}_q) / B_n$ we know that $$[n]! = \sum_{ \sigma \in S(n)} q^{l(\sigma)}$$ where $[n]! = \prod_{j=1}^n (1+q + \cdots + q^{j-1})$ and $l(\sigma)$ is the length of the permutation $\sigma \in S(n)$ (also known as the number of involutions of $\sigma$).</p> <p>We also know that $$\theta^{(n)} = \sum_{\sigma \in S(n)} \theta^{[\sigma]}$$ where $[\sigma]$ is the number of cycles of the permutation $\sigma \in S(n)$ and $\theta^{(n)} = \theta(\theta+1) \cdots (\theta+n-1)$. Notice that $$\lim_{q \rightarrow 1} \ [n]! = n! = \lim_{\theta \rightarrow 1} \ \theta^{(n)}.$$ Is there a way to write $$\sum_{\sigma \in S(n)} q^{l (\sigma)} \theta^{[\sigma]}$$ explicitly as a function of $q$ and $\theta$?</p> http://mathoverflow.net/questions/68121/infinitely-divisible-distributions-and-maximal-entropy Infinitely Divisible Distributions and Maximal Entropy Alexander Moll 2011-06-18T06:54:22Z 2011-08-26T14:14:21Z <p>The normal distribution on $\mathbb{R}$, the exponential distribution on $\mathbb{R}_{\geq 0}$, and the geometric distribution on $\mathbb{N}$ are examples of distributions that are both infinitely divisible and entropy maximizers. On the other hand, the Poisson distribution is an infinitely divisible distribution on $\mathbb{N}$ without maximizing entropy, while the uniform distribution on the interval $[a,b]$ maximizes entropy but is not infinitely divisible.</p> <p>Can anything be said about the relationship between these two classes of distributions?</p> http://mathoverflow.net/questions/70586/sums-of-partitions-and-stirlings-formula Sums of Partitions and Stirling's formula Alexander Moll 2011-07-18T01:08:20Z 2011-08-02T05:20:19Z <p>Stirling's formula $$N! \sim \sqrt{2 \pi}\ N^{N+ \frac{1}{2}} e^{-N}$$ follows easily from Laplace's method in light of the famous integral representation $$N! = \int_0^{\infty} e^{-z} z^N dz.$$ Basic representation theory of the symmetric group $S(N)$ gives the remarkable finite sum $$N! = \sum_{\lambda \in \mathbb{Y}_N} (\dim \lambda)^2$$ over all partitions of $N$, where $\dim \lambda$ counts the number of paths from $\emptyset$ to $\lambda$ in Young's lattice $\mathbb{Y}$.</p> <ul> <li>Is it possible to derive Stirling's formula directly from this finite sum?</li> </ul> <p> I'd like to thank everyone for the very interesting comments below. For those looking at this thread for the first time, I was hoping that an answer to the question above might help us calculate the asymptotics of $$f(N,a) = \sum_{\lambda \in \mathbb{Y}_N} (\dim \lambda)^{a}.$$ Notice that the case $a=0$ is the partition function $|\mathbb{Y}_N|$.</p> http://mathoverflow.net/questions/70710/continuous-measurement-in-quantum-mechanics Continuous Measurement in Quantum Mechanics Alexander Moll 2011-07-19T05:12:40Z 2011-08-01T18:58:51Z <p>Let $\mathcal{P}(S^{\infty})$ denote the set of probability measures on the unit sphere $S^{\infty} \subset \mathcal{H}$ in the Hilbert space of states of a quantum mechanical system. Measurement of an observable $\Omega$ corresponds to orthogonal projection, sending $\delta_{|\psi \rangle}$ to a particular measure supported on the eigenvectors of $\Omega$, thus inducing a map $T_{\Omega}: \mathcal{P}(S^{\infty}) \rightarrow \mathcal{P}(S^{\infty})$. If we think of $T_{\Omega}$ as the transition matrix of a Markov chain, we can say that a continuum $\lbrace \Omega_t \rbrace_{t \in \mathbb{R} \geq 0}$ of observables induces a stochastic process on $S^{\infty}$.</p> <ul> <li><p>If we let $\Omega_t = H(t)$, the time-dependent Hamiltonian of our system, is the associated stochastic process the deterministic one described by the Schrödinger equation?</p></li> <li><p>What does this construction have to do with the time-energy uncertainty relation?</p></li> </ul> http://mathoverflow.net/questions/69380/probabilistic-solution-of-the-porous-medium-equation Probabilistic Solution of the Porous Medium Equation Alexander Moll 2011-07-03T04:17:58Z 2011-07-18T13:41:58Z <p>It is well known that the transition density for standard Brownian motion $B_t$ in $\mathbb{R}^d$ yields a solution to the global Cauchy problem for the heat equation $$u_t = \Delta u$$ with initial condition given by the Dirac distribution $\delta_0$.</p> <p>Unlike the heat equation, the porous medium equation $$u_t = \Delta(u^m)$$ with exponent $m>1$ has finite speed of propagation.</p> <ul> <li><p>When is the infinitesimal generator of a stochastic process linear?</p></li> <li><p>Is there a probabilistic solution of this non-linear diffusion equation?</p></li> </ul> http://mathoverflow.net/questions/67003/least-squares-regression-and-differential-geometry Least-squares regression and differential geometry Alexander Moll 2011-06-05T23:56:30Z 2011-06-06T14:57:33Z <p>For $k, n \in \mathbb{N}$, let $\mathcal{C}_n \mathbb{R}^k$ denote the configuration space of $n$ distinct points in $\mathbb{R}^k$. </p> <ul> <li>(1) Is there a description of the tangent space $T_C \mathcal{C}_n \mathbb{R}^k$ in terms of the configuration $C$?</li> </ul> <p>Equipping $\mathbb{R}^k$ with the usual metric, a regression line $l_C$ of a collection $C \in \mathcal{C}_n \mathbb{R}^k$ is a line minimizing the quantity $E_C = \sum\limits_{p \in C} d(p, l_C)^2.$ We can see this as variational problem $E_C: M^k \rightarrow \mathbb{R}$ where $M^k$ is the parameter space of all lines in $\mathbb{R}^k$. </p> <ul> <li>(2) Is there an explicit parametrization of $M^k$? </li> </ul> <p>Without this knowledge, I'm not sure how to proceed to check whether $E_C$ is a Morse function.</p> <p>[Note for $k=2$: given $C \in \mathcal{C}_n \mathbb{R}^k$, since $n&lt;\infty$ we can always find an angle $\theta$ such that a rotation of our axes by $\theta$ yields coordinates $(x,y)$ on $\mathbb{R}^2$ for which the $x$-values of the $p \in C$ are all distinct, bringing us back to function-fitting and the usual least-squares regression which minimizes only the distances in the $y$ direction.]</p> http://mathoverflow.net/questions/59443/on-the-decomposition-of-two-representations-of-the-iwahori-hecke-algebra-of-type On the decomposition of two representations of the Iwahori-Hecke algebra of type A_n Alexander Moll 2011-03-24T16:00:47Z 2011-03-24T16:00:47Z <p>Consider the Iwahori Hecke algebra $H_q(n)$ of the symmetric group $S(n)$, the $\Bbb{Z}[q^{1/2},q^{-1/2}]$ algebra with generators $T_i$, $1 \leq i \leq n-1$, subject to the braid relations and the quadratic relation $T_i^2 = (q-1)T_i + q$.</p> <p>(i) Let $V_k$ be the defining representation of the quantum super-group $U_{q,k,m}:=U_q(\mathfrak{gl}(k+m|m))$. The commutant of the diagonal action of $U_{q,k,m}$ on $V_k^{\otimes n}$ is generated by the representation of $H_q(n)$ on this space $\forall k, m \in \Bbb{N}$.</p> <p>(ii) Consider the left action of the group of invertible $n \times n$ matrices $GL(n, q)$ with coefficients in $\mathbb{F}_q$ on the full flag variety $X(n) := GL(n,q) / B$ and take the corresponding representation in the vector space $L[X(n)]$ of functions (or measures) on $X(n)$. The space of intertwiners of this representation of $GL(n,q)$ is isomorphic to $H_q(n)$.</p> <p>Q1) How does the representation in (ii) decompose into irreducible representations of $GL(n,q)$?</p> <p>Q2) Is there any relationship between $GL(n,q)$ and $U_{q,k,m}$?</p> http://mathoverflow.net/questions/30428/on-the-cohomology-ring-of-the-grassmannian On the cohomology ring of the Grassmannian Alexander Moll 2010-07-03T16:38:44Z 2010-07-06T03:07:15Z <p>The basis of Schubert classes for the cohomology ring $H^*(\text{Gr}(m,N))$ of the Grassmannian of $m$-dimensional subspaces of $\mathbb{C}^N$ is indexed by $L(m,N-m)$, the poset of all partitions fitting inside a $m \times (N-m)$ box. This is the quotient of the powerset $2^{m(N-m)}$ by the action of the wreath product $S(m) \wr S(N-m)$. How does this come from the fact that $\text{Gr}(m,N) \cong U(N) / \big (U(m) \times U(N-m) \big )$? Can this be extended to other homogeneous spaces?</p> http://mathoverflow.net/questions/124923/evaluating-an-infinite-product-of-q-exponentials Comment by Alexander Moll Alexander Moll 2013-03-19T15:28:06Z 2013-03-19T15:28:06Z Hey Ben - use the last comment in <a href="http://en.wikipedia.org/wiki/Q-exponential" rel="nofollow">en.wikipedia.org/wiki/Q-exponential</a> http://mathoverflow.net/questions/124923/evaluating-an-infinite-product-of-q-exponentials Comment by Alexander Moll Alexander Moll 2013-03-19T03:32:46Z 2013-03-19T03:32:46Z Hopefully something else - but you're right that for a certain value of $x$ as a function of $q$ this will return the MacMahon generating function for plane partitions. http://mathoverflow.net/questions/124586/is-there-a-sequence-of-finite-groups Comment by Alexander Moll Alexander Moll 2013-03-15T04:12:26Z 2013-03-15T04:12:26Z Oh good point! It seems I didn't scroll down far enough when I searched &quot;plane partitions&quot; within MO. http://mathoverflow.net/questions/123032/two-appearances-of-the-jacobi-triple-product-identity Comment by Alexander Moll Alexander Moll 2013-02-26T23:05:23Z 2013-02-26T23:05:23Z Also, I'm hoping to use this observation as an opportunity to start investigating such papers - I'm just not sure where to begin! http://mathoverflow.net/questions/123032/two-appearances-of-the-jacobi-triple-product-identity Comment by Alexander Moll Alexander Moll 2013-02-26T22:57:16Z 2013-02-26T22:57:16Z The second approach is essentially the proof given in <a href="http://en.wikipedia.org/wiki/Jacobi_triple_product" rel="nofollow">en.wikipedia.org/wiki/Jacobi_triple_product</a> http://mathoverflow.net/questions/92736/two-curious-asymptotic-results-for-dimensions-of-type-a-objects/93005#93005 Comment by Alexander Moll Alexander Moll 2012-04-03T22:27:03Z 2012-04-03T22:27:03Z This is a very nice argument Hugh - thank you! http://mathoverflow.net/questions/92736/two-curious-asymptotic-results-for-dimensions-of-type-a-objects/92772#92772 Comment by Alexander Moll Alexander Moll 2012-04-01T00:16:51Z 2012-04-01T00:16:51Z i.e. is there a bijective'' proof of the limit above? http://mathoverflow.net/questions/92736/two-curious-asymptotic-results-for-dimensions-of-type-a-objects/92772#92772 Comment by Alexander Moll Alexander Moll 2012-04-01T00:16:16Z 2012-04-01T00:16:16Z Sorry - I'm familiar with this approach to calculating the limit, and should have included it in my original post, where I was aiming for more of a representation theoretic &quot;Why?&quot;. Maybe we can ask this question combinatorially: is there a reason why the number of semi standard Young tableaux of shape $\lambda$ with entries from $\{1, \ldots, N\}$ after dividing by $N^n$ is asymptotic to the number of standard Young tableaux with shape $\lambda$ if we divide by $n!$? http://mathoverflow.net/questions/92736/two-curious-asymptotic-results-for-dimensions-of-type-a-objects Comment by Alexander Moll Alexander Moll 2012-03-31T07:09:59Z 2012-03-31T07:09:59Z Also, something I really want is some representation theoretic place where I might find a direct sum of $W_{\lambda, N} \boxtimes W_{\lambda, M}$ over all $\lambda \in \mathbb{Y}$ with length $l(\lambda) \leq \min (N,M)$ - a sort of unbalanced'' Peter-Weyl theorem for $SU(N)$ on the left and $SU(M)$ on the right. Any thoughts? http://mathoverflow.net/questions/81221/graduate-ode-textbook Comment by Alexander Moll Alexander Moll 2011-11-18T05:42:45Z 2011-11-18T05:42:45Z I like Prof. Teschl's notes <a href="http://www.mat.univie.ac.at/~gerald/ftp/book-ode/index.html" rel="nofollow">mat.univie.ac.at/~gerald/ftp/book-ode/index.html</a> http://mathoverflow.net/questions/81035/coincidences-amongst-classifying-spaces-and-eilenberg-mac-lane-spaces/81045#81045 Comment by Alexander Moll Alexander Moll 2011-11-16T15:35:54Z 2011-11-16T15:35:54Z Thanks for the reference, Mark - that book looks really cool! http://mathoverflow.net/questions/81035/coincidences-amongst-classifying-spaces-and-eilenberg-mac-lane-spaces/81045#81045 Comment by Alexander Moll Alexander Moll 2011-11-16T15:21:27Z 2011-11-16T15:21:27Z My first reaction was Tannaka-Krein dualtiy, but I feel like there should be something more concrete for SU(2), probably its Langlands dual group since it's reductive. I'd be surprised if there wasn't any pre-existing work on/near this topic... http://mathoverflow.net/questions/81035/coincidences-amongst-classifying-spaces-and-eilenberg-mac-lane-spaces/81045#81045 Comment by Alexander Moll Alexander Moll 2011-11-16T14:18:43Z 2011-11-16T14:18:43Z Thanks for the response - I'm certainly glad to see a 4. I'm not familiar with rational homotopy theory, but I'd like to ask: is there some way of understanding the rationalization of BG in terms of the representation theory of G? How can one explain this $\mathbb{Z}$ in the result for $BSp(1)$? http://mathoverflow.net/questions/2899/computations-in-knot-homology-theories Comment by Alexander Moll Alexander Moll 2011-11-16T07:24:49Z 2011-11-16T07:24:49Z Hi - though I'm no expert, the paper of Carqueville and Murfet <a href="http://arxiv.org/abs/1108.1081" rel="nofollow">arxiv.org/abs/1108.1081</a> seems relevant. http://mathoverflow.net/questions/81035/coincidences-amongst-classifying-spaces-and-eilenberg-mac-lane-spaces Comment by Alexander Moll Alexander Moll 2011-11-16T06:32:24Z 2011-11-16T06:32:24Z $\widehat{G}$ denotes the dual group of a locally compact abelian group. In particular, $\widehat{G} \cong G$ for finite groups (but not canonically), and you may recall $\widehat{U(1)} \cong \mathbb{Z}$ from Fourier series. For a non-abelian group, the machinery of K(G,n)s cannot possibly work, which is the point of the question.