Concrete Nonsense » Representation Theory http://concretenonsense.wordpress.com A group blog about mathematics Mon, 30 Nov 2009 21:48:46 +0000 http://wordpress.com/ en hourly 1 http://www.gravatar.com/blavatar/2b77f97368a020180b21c93e224b3ae1?s=96&d=http://s.wordpress.com/i/buttonw-com.png Concrete Nonsense » Representation Theory http://concretenonsense.wordpress.com GLFq III: characteristic map http://concretenonsense.wordpress.com/2009/10/12/glfq-iii-characteristic-map/ http://concretenonsense.wordpress.com/2009/10/12/glfq-iii-characteristic-map/#comments Mon, 12 Oct 2009 15:47:33 +0000 Steven Sam http://concretenonsense.wordpress.com/?p=710 ]]>

In the last post of this series, I gave some definitions and facts regarding the Hall–Littlewood functions. I also sketched the relationship between symmetric functions and representations of the symmetric group. Now we’ll see how this works for the finite general linear groups.

We want to imitate the Frobenius character that is used to relate the characters of the symmetric group to the ring of symmetric functions. But since the description of the conjugacy classes of the finite general linear group (and hence the parametrization of its irreducible characters) are more complicated than the description for the symmetric group, we’ll need a bigger ring to work with.

We continue the notation from the first post. For each irreducible polynomial f \in \Phi and each positive integer i>0, we have a variable X_{i,f}, whose degree we set to be deg(f). For any symmetric function u, we set u(X_f) = u(X_{1,f}, X_{2,f}, \dots). The graded ring we work in is B = {\bf C}[e_n(X_f) \mid n \ge 1,\ f \in \Phi], where e_n denotes the elementary symmetric function. In other words, elements of B are functions which are symmetric in each family of variables X_f.

Recall from last time that for a partition \lambda, P_\lambda(x;t) and Q_\lambda(x;t) are the Hall–Littlewood and augmented Hall–Littlewood functions. We use these to define elements in B:

\tilde{P}_\lambda(X_f) = q^{-\deg(f) n(\lambda)} P_\lambda(X_f; q^{-\deg(f)})

\tilde{Q}_\lambda(X_f) = q^{\deg(f)(|\lambda| + n(\lambda))} Q_\lambda(X_f; q^{-\deg(f)}),

where n(\lambda) = \sum_i (i-1)\lambda_i. For a partition-valued function \boldsymbol{\mu}, we set

\displaystyle \tilde{P}_{\boldsymbol{\mu}} = \prod_{f \in \Phi} \tilde{P}_{\boldsymbol{\mu}(f)}(X_f),

\displaystyle \tilde{Q}_{\boldsymbol{\mu}} = \prod_{f \in \Phi} \tilde{Q}_{\boldsymbol{\mu}(f)}(X_f).

We use these two bases to define a (complex) inner product on B:

\langle \tilde{P}_{\boldsymbol{\lambda}}, \tilde{Q}_{\boldsymbol{\mu}} \rangle = \delta_{\boldsymbol{\lambda}, \boldsymbol{\mu}}.

Now we need to construct the representation ring of the finite general linear groups G_n over {\bf F}_q, where now q will remain fixed. This will be very similar to what happens for the symmetric groups. Given characters u and v for G_n and G_m, respectively, let P be the parabolic subgroup of G_{n+m} consisting of matrices of the form

g(A,B,C) = \begin{bmatrix} A & B \\ 0 & C \end{bmatrix}

where A is an n \times n matrix, B is an n \times m matrix, and C is an m \times m matrix. We define a character w on P by setting

w(g(A,B,C)) = u(A) v(B).

Then the induction product u \circ v is defined as the induced character {\rm Ind}_P^{G_{n+m}}(w). (Recall that for symmetric groups, we define the induction product by inducing from parabolic subgroups \mathfrak{S}_n \times \mathfrak{S}_m \subset \mathfrak{S}_{n+m}.) If we let A_n denote the complex vector space of characters of G_n, then the induction product gives a graded ring structure on A = \bigoplus_{n \ge 0} A_n. We can also put a complex inner product on A by setting the different graded components to be mutually orthogonal and using the standard inner product for characters on each component, just as in the case of the symmetric group. Now comes the important part: let \pi_{\boldsymbol{\mu}} denote the function which is 1 on the conjugacy class corresponding to \boldsymbol{\mu}, and 0 elsewhere. Then we have a characteristic map

{\rm ch} \colon A \to B, \quad {\rm ch}(\pi_{\boldsymbol{\mu}}) = \tilde{P}_{\boldsymbol{\mu}}.

Theorem. The characteristic map ch is an isometric isomorphism of graded rings.

If we continue with the analogy of the relationship between the symmetric group and symmetric functions, then the characteristic of the irreducible characters of G_n should be some kind of “Schur functions.” Unfortunately their definition will require significantly more notation. So I’ll skip that and just say that we can define functions S_{\boldsymbol{\lambda}}. One catch, though, is that the indexing set we use for these Schur functions is not the same as the indexing set for conjugacy classes. The indices {\boldsymbol{\lambda}} can be thought of as partition-valued functions, but on a different domain. But this is not such a big deal.

Theorem. The S_{\boldsymbol{\lambda}} form an orthonormal basis for B. Furthermore, their inverses under the characteristic map are the irreducible characters \chi^{\boldsymbol{\lambda}} of the groups G_n. Consequently, the values of the characters are given by the change of bases S_{\boldsymbol{\lambda}} = \sum_{\boldsymbol{\mu}} \chi^{\boldsymbol{\lambda}}(\boldsymbol{\mu}) \tilde{P}_{\boldsymbol{\mu}}.

At any rate, I think it is nice that the same kind of setup works for the finite general linear groups as does for the symmetric group, which maybe further justifies the statement that the finite general linear groups are q-analogues of the symmetric groups.

But since these symmetric functions are so horribly complicated, one doesn’t expect to have a nice combinatorial rule for changing from the S basis to the \tilde{P} basis (such as the Murnaghan–Nakayama rule for writing the Schur polynomials in terms of power sum symmetric functions in the symmetric group case). There are some nice cases though. When the conjugacy class \boldsymbol{\mu} corresponds to a unipotent conjugacy class, we can evaluate induced characters from maximal tori T of G_n at \boldsymbol{\mu} to get Green polynomials (up to a sign). And sometimes these induced characters are irreducible (precisely when the stabilizer of the character in the Weyl group of T is trivial).

Green polynomials are more manageable to think about: they arise as the change of basis coefficients when writing power sum symmetric functions as Hall–Littlewood functions (now working just in the ring \Lambda[t] from last time).

That’s basically all I want to say about the connection between symmetric functions and finite general linear groups. There is a more powerful approach to characters of these groups using \ell-adic cohomology due to Deligne and Lusztig, and it works more generally for any finite group of Lie type. Using that approach, it can be shown, for example, that the characters are integer valued.

-Steven

]]> http://concretenonsense.wordpress.com/2009/10/12/glfq-iii-characteristic-map/feed/ 0 masnevets GLFq II: Hall–Littlewood functions http://concretenonsense.wordpress.com/2009/09/28/glfq-ii-hall-littlewood-functions/ http://concretenonsense.wordpress.com/2009/09/28/glfq-ii-hall-littlewood-functions/#comments Mon, 28 Sep 2009 14:21:27 +0000 Steven Sam http://concretenonsense.wordpress.com/?p=684 ]]>

Last time, I discussed how to parametrize the conjugacy classes of the general linear group G over a finite field. In a sense, this group is a q-analogue of the symmetric group, so we might try to imitate the constructions for the symmetric group to get information about G. There’s a nice construction that Frobenius worked out which connects the characters of the symmetric group with the combinatorics of the Schur functions. I’ll briefly summarize the statement. The conjugacy classes of the symmetric group on n letters are parametrized by partitions of n. So we can also parametrize the irreducible characters by partitions as well (though it is not clear how to do this in a “canonical” way a priori). Ignoring the indexing issue (which can be dealt with) and letting \chi^\lambda(\mu) be the irreducible character indexed by \lambda evaluated at the conjugacy class consisting of permutations whose cycle lengths are given by the parts of \mu, then one has s_\lambda(x) = \sum_\mu z_\mu^{-1} \chi^\lambda(\mu) p_\mu(x) where s_\lambda(x) is a Schur function, p_\mu(x) is a power sum (Newton) symmetric function, and z_\mu = 1^{m_1} m_1! 2^{m_2} m_2! \cdots where m_i is the number of times that i appears as a part of \mu (the meaning of z_\mu is that n! z_\mu^{-1} is the size of the conjugacy class index by \mu.)

So the question to ask might be “can we find a similar interpretation for the characters of G?” The answer is yes, but becomes a bit more involved.

Instead of Schur functions, one needs to look at another class of symmetric functions called the Hall–Littlewood functions, but we’ll actually need a much larger ring than the ring of symmetric functions. While the Schur functions are symmetric functions in a set of variables x_1, x_2, \dots usually defined over rational coefficients, the Hall–Littlewood (HL) functions are symmetric functions defined over the ring {\bf Z}[t] (so t is an additional variable which does not affect the definition of “symmetric”.) Let \Lambda[t] denote the symmetric functions in variables x_1, x_2, \dots with coefficients in {\bf Z}[t]. Like the Schur functions, the HL functions are indexed by partitions. The definition of the Hall–Littlewood function indexed by \lambda in n variables x_1, \dots, x_n is

\displaystyle P_\lambda(x_1, \dots, x_n; t) = \sum_{w \in S_n / S_n^\lambda} w\left( x_1^{\lambda_1} \cdots x_n^{\lambda_n} \prod_{\lambda_i > \lambda_j} \frac{x_i - tx_j}{x_i - x_j} \right) .

Here S_n is the symmetric group on n letters, and S_n^\lambda \subseteq S_n is the subgroup of permutations w such that \lambda_{w(i)} = \lambda_i for all i. From this, one can see that P_\lambda(x_1, \dots, x_n; 1) = m_\lambda(x_1, \dots, x_n) where m_\lambda denotes the monomial symmetric function which is the sum of all of the distinct terms x_{w(1)}^{\lambda_1} \cdots x_{w(n)}^{\lambda_n} as w ranges over all permutations of n.

Set [\lambda]_t! = [\lambda_1]_t! \cdots [\lambda_n]_t!. An equivalent definition for P_\lambda is

\displaystyle P_\lambda(x_1, \dots, x_n; t) = [\lambda]_t!^{-1} \sum_{w \in S_n} w \left( x_1^{\lambda_1} \cdots x_n^{\lambda_n} \prod_{i < j} \frac{x_i - tx_j}{x_i - x_j} \right),

from which one can deduce that P_\lambda(x_1, \dots, x_n; 0) = s_\lambda(x_1, \dots, x_n) from the Weyl character formula.

I just want to state some of the properties that we will need later without giving too many details. For proofs, one can consult Macdonald's book Symmetric Functions and Hall Polynomials. From the first definition, one can deduce that these functions enjoy a stability property:

P_\lambda(x_1, \dots, x_n, 0; t) = P_\lambda(x_1, \dots, x_n; t),

and hence one can define P_\lambda(x;t) in infinitely many variables x_1, x_2, \dots by taking an inverse limit. Since the P_\lambda are symmetric functions, we can write

P_\lambda(x;t) = \sum_\mu w_{\lambda, \mu} (t) s_\mu(x)

for some polynomials w_{\lambda,\mu}(t). In fact, w_{\lambda, \lambda}(t) = 1 and w_{\lambda,\mu}(t) = 0 unless \lambda \ge \mu (dominance order), so the change of basis matrix from P_\lambda to s_\lambda is upper unitriangular, which implies that the P_\lambda(x;t) form a {\bf Z}[t]-basis of \Lambda[t].

The inverse of this change of basis is very interesting. In this case, write s_\lambda(x) = \sum_\mu K_{\lambda, \mu}(t) P_\mu(x;t). The K_{\lambda, \mu}(t) are the Kostka–Foulkes polynomials. Since P_\mu(x;1) = m_\mu(x), we see that K_{\lambda, \mu}(t) = K_{\lambda, \mu} are the Kostka numbers. It is a fact that the Kostka–Foulkes polynomials are in fact polynomials, and they have nonnegative integers. I hope to write a post about these at some point.

We will also need augmentations of these functions in the next post. First set

\displaystyle b_\lambda(t) = \prod_{i \ge 1} [(1-t)(1-t^2) \cdots (1-t^{m_i(\lambda)})]

where m_i(\lambda) is the multiplicity with which i appears in \lambda. Then define

Q_\lambda(x; t) = b_\lambda(t) P_\lambda(x; t).

Although we won’t use them, let me mention skew Hall–Littlewood functions. Define an inner product on \Lambda[t] by declaring that \langle P_\lambda(x;t), Q_\mu(x;t) \rangle = \delta_{\lambda, \mu}. Then we can define skew Hall–Littlewood functions for partitions \mu \subseteq \lambda via

\langle Q_{\lambda/\mu}(x;t), P_\nu(x;t) \rangle = \langle Q_\lambda(x;t), P_\mu(x;t) P_\nu(x;t) \rangle

\langle P_{\lambda/\mu}(x;t), Q_\nu(x;t) \rangle = \langle P_\lambda(x;t), Q_\mu(x;t) Q_\nu(x;t) \rangle.

From this definition, setting t=0 gives back the skew Schur functions s_{\lambda/\mu}(x) (since they are defined in a similar way). The weird thing, however, is that the skew Schur functions only depend on the shape \lambda/\mu, whereas the skew Hall–Littlewood functions remember both \lambda and \mu. One can write down a rather explicit formula for Q_{\lambda/\mu} in terms of semistandard tableaux which shows that the function depends on both \lambda and \mu (but this is only seen in the powers of t, and not the x_i), but I will omit this so that I can wrap this post up.

Let me just end with some other specializations of t that are important. When \lambda is a strict partition (i.e., the nonzero parts are distinct) then setting t=-1 gives P_\lambda(x;-1) = 2^{\ell(\lambda)} Q_\lambda where \ell(\lambda) is the number of parts, and Q_\lambda are the Schur Q-functions, which are important for the projective representation theory of the symmetric group (maybe a future topic). Also, specializations at t=q^{-1} for q a prime power are related to Hall algebras, which are used to keep track of extensions between finite Abelian groups.

In the next post, I’ll discuss the connection between characters of {\bf GL}_n({\bf F}_q) and symmetric functions.

-Steven

]]> http://concretenonsense.wordpress.com/2009/09/28/glfq-ii-hall-littlewood-functions/feed/ 0 masnevets GLFq I: Conjugacy classes of a finite general linear group http://concretenonsense.wordpress.com/2009/09/14/glfq-i-conjugacy-classes-of-a-finite-general-linear-group/ http://concretenonsense.wordpress.com/2009/09/14/glfq-i-conjugacy-classes-of-a-finite-general-linear-group/#comments Mon, 14 Sep 2009 15:26:32 +0000 Steven Sam http://concretenonsense.wordpress.com/?p=665 ]]>

I’d like to start a series of posts, GLFq (general linear group over a finite field), in which I hope to explain the representation theory of G = {\bf GL}_n({\bf F}_q) (n and q will remain fixed). In this first post, I’ll explain how to write down the conjugacy classes of G. In later posts, I plan to introduce Hall–Littlewood polynomials and the characteristic map. I would like to also go into how to construct the actual representations, and discuss things related to Hall–Littlewood polynomials, like the q-Kostka polynomials and a lot of the interesting algebra/geometry behind them.

There are two pieces of data we would like to know. First, what is the size of G? Second, how do we parameterize the conjugacy classes? The first question is easy to answer since an invertible matrix is given by the data of n linearly independent vectors. The first one can be chosen to be any nonzero vector, so there are q^n - 1 of them. In general, the ith one can be chosen to be any vector not in the span of the last i-1 (so we are just avoiding some i-1 dimensional subspace, which has q^{i-1} elements), and hence there are q^n - q^{i-1} choices for such a vector. All together, the number of elements of G is \prod_{i=1}^n (q^n-q^{i-1}). We can rewrite this as q^{\binom{n}{2}} (q-1)^n [n]_q! to make it more analogous to the number of elements of the symmetric group.

The second question requires the rational canonical form. If we were dealing with an algebraically closed field, conjugacy classes would of course be parameterized by Jordan normal forms, so we need some kind of substitute for that. First, we need the structure theorem for finitely generated modules over a principal ideal domain R. This says that any such module is a direct sum of its torsion submodule and a free submodule. Furthermore, the torsion submodule is uniquely a direct sum of cyclic modules, which can be written in the form R/(f^m) for some irreducible element f and some positive integer m.

Given a matrix A, we’ll apply this to the case R = {\bf F}_q[t] and the module V = {\bf F}_q^n where the action of a polynomial p(t) on V is given by p(A). Irreducible elements of R are the same as irreducible polynomials, but we will never see the polynomial x show up if A is invertible, and we will only need to use the monic ones. Let \Phi be the set of all monic irreducible polynomials over {\bf F}_q which are different from the constant polynomial x. Hence, we see that the data of the decomposition is given by a partition valued function \boldsymbol{\mu} on \Phi. Explicitly, the function \boldsymbol{\mu} gives the module \bigoplus_{f \in \Phi} \bigoplus_{i \ge 0} R / (f^{\boldsymbol{\mu}(f)_i}).

Since the dimension of R / (f^m) is \deg(f) \cdot m, the conjugacy classes of G are given by those partition valued functions \boldsymbol{\mu} such that \| \boldsymbol{\mu} \| := \sum_{f \in \Phi} \sum_i \deg(f) \boldsymbol{\mu}(f)_i = n.

As an aside, if we look at the conjugacy classes of all matrices instead of invertible matrices, i.e., we allow the domain of the partition valued functions above to include the polynomial x, then the number of such classes is \sum_j p_j(n) q^j where p_j(n) is the number of partitions of n into j parts. I think it’s a really nice formula (though it takes some work to show). See Chapter 1, Section 10 of the second edition of Enumerative Combinatorics I for a derivation of this formula.

Let’s look at the case of n=2. The only valid partition valued functions can only have nonempty values on polynomials of degree at most 2. There are 3 types of functions:

  • There exists a single monic irreducible polynomial f of degree 1 such that \mu = \boldsymbol{\mu}(f) \ne \emptyset and we have \sum_i \mu_i = 2, so \mu \in \{(2), (1,1)\}. These correspond to matrices with a single eigenvalue, the partition (2) means that it’s a diagonal matrix, and the partition (1,1) means that it is conjugate to a size 2 Jordan block.
  • There exists two distinct monic irreducible polynomials f and g of degree 1 such that \boldsymbol{\mu} takes the value (1) on both and is empty on all other polynomials. These correspond to matrices with two distinct eigenvalues.
  • There exists a single monic irreducible polynomial f of degree 2 such that \boldsymbol{\mu}(f) = (1) and all other values are the empty partition. These are matrices without a Jordan normal form.

The only irreducible polynomials of degree 1 which are allowed are of the form x-a for a nonzero value of a. So there are 2(q-1) functions of the first kind and \binom{q-1}{2} = (q-1)(q-2)/2 functions of the second kind. For the third kind, we have q^2 monic polynomials of degree 2 over {\bf F}_q. There are q polynomials of the form (x-a)^2 and \binom{q}{2} = q(q-1)/2 of the form (x-a)(x-b) for a and b distinct, so we must have q^2 - q - q(q-1)/2 = q(q-1)/2 monic irreducible degree 2 polynomials over {\bf F}_q. Thus in total we have 2(q-1) + (q-1)(q-2)/2 + q(q-1)/2 = q^2 - 1 conjugacy classes.

Next time, I’ll say something about Hall–Littlewood polynomials. Click here for the next post.

-Steven

]]> http://concretenonsense.wordpress.com/2009/09/14/glfq-i-conjugacy-classes-of-a-finite-general-linear-group/feed/ 0 masnevets A Fock space representation http://concretenonsense.wordpress.com/2009/08/31/a-fock-space-representation/ http://concretenonsense.wordpress.com/2009/08/31/a-fock-space-representation/#comments Mon, 31 Aug 2009 13:39:23 +0000 Steven Sam http://concretenonsense.wordpress.com/?p=623 ]]>

Lately I’ve been reading about quantum groups. I particularly enjoyed Bernard Leclerc’s paper Symmetric functions and the Fock space representation of U_q(\widehat{\mathfrak{sl}_n}), so I want to discuss a little bit of it. In particular, I want to give an example of the technique of deforming an object in order to see “hidden structure.” My notation will differ slightly from Leclerc’s since he uses the French notation for Young diagrams.

One thing that has always been really hard for me to wrap my head around is the really complicated presentations that affine Lie algebras have and how one is supposed to do anything with them. This post will be about the affine Lie algebra \widehat{\mathfrak{sl}_n}, which is defined in the paper as the Lie algebra with generators e_i, f_i, h_i (0 \le i \le n-1) and d, with 5 lines of relations. Let K be a field of characteristic 0. Another way around this is to first define the loop algebra L(\mathfrak{g}) of a simple Lie algebra \mathfrak{g} as: L(\mathfrak{g}) = \mathfrak{g} \otimes K[t,t^{-1}] with a Lie bracket given by [a \otimes t^n, b \otimes t^m] = [a,b]_{\mathfrak{g}} \otimes t^{n+m}, and then to say that \widehat{\mathfrak{g}} is its universal central extension. More precisely, we say add a central element c, and then extend the bracket above via [a \otimes t^n, b \otimes t^m] = [a,b]_{\mathfrak{g}} \otimes t^{n+m} + (a,b)_{\mathfrak{g}} n \delta_{n,-m} c where \delta is the Kronecker delta, and (,) is the Killing form of \mathfrak{g}.

In the case that \mathfrak{g} is \mathfrak{sl}_n, I want to discuss a more concrete (combinatorial) description. Just as \mathfrak{sl}_n can be thought as the traceless operators on an n-dimensional vector space (the standard representation), we can also find a standard representation for \widehat{\mathfrak{sl}_n} (the Fock space representation). For this, let {\cal F} denote the ring of symmetric functions over K in infinitely many variables. The Schur functions s_\lambda form a basis indexed by partitions, and this will be our representation. In order to describe the actions of \widehat{\mathfrak{sl}_n} on Sym, we’ll need some notation.

First, we represent partitions \lambda by their Young diagram (\lambda_i boxes drawn in the ith row, left justified). The content of a box (i,j) is the number i-j. We’ll consider contents modulo n, and say that a box is an i-node if it has content i modulo n. We define e_is_\lambda = \sum_\mu s_\mu (resp. f_is_\lambda = \sum_\mu s_\mu) where the sum is over all \mu obtained from \lambda by removing (resp. adding) an i-node, and define ds_\lambda = N_0(\lambda)s_\lambda where N_0(\lambda) is the number of 0-nodes of \lambda. Finally, set h_i = e_if_i - f_ie_i. Then \widehat{\mathfrak{sl}_n} is the Lie algebra spanned by these generators.

Unlike the \mathfrak{sl}_n case, the Fock space representation is not irreducible. Let p_\lambda = \prod_i p_{\lambda_i} be the power sum symmetric function where p_i = \sum_{j \ge 1} x_j^i. It turns out that the set \{p_{n\lambda}\} where n\lambda = (n\lambda_1, n\lambda_2, \dots) are the highest weight vectors of this representation (i.e., they are killed by the e_i, and are eigenvectors for d and the h_i). Furthermore, one has dp_{n\lambda} = |\lambda| p_{n\lambda}, so we have a natural notion of degree for our highest weight vectors. Within these graded subsets, the p_{n\lambda} give an obvious choice of basis, but there is no reason to favor them: for example, \{ p_{(2n)} + p_{(n,n)}, p_{(2n)} - p_{(n,n)} \} also forms a basis for the highest weight vectors of degree 2. The point is that the Schur functions give a “natural basis” for {\cal F} in the sense that we have defined our operators in this basis, and the basis of highest weight vectors should have “nice” properties with respect to this fixed basis, although it’s not clear what nice means right now.

The next part is something that I am learning to appreciate: since there is no way to figure out a “canonical basis” for the highest weight vectors, we should introduce a new parameter to make the structure of the Fock space representation more rigid. This new parameter is made precise by replacing \widehat{\mathfrak{sl}_n} by its q-analogue U_q(\widehat{\mathfrak{sl}_n}), and similarly for {\cal F}. More precisely, we can’t deform the Lie algebra, but we can deform its universal enveloping algebra. The quantum group U_q(\widehat{\mathfrak{sl}_n}) has generators E_i, F_i, K_i, K_i^{-1}, and D, D^{-1} and even more relations than \widehat{\mathfrak{sl}_n} has, so rather than give those, I just want to mention how to change the action on {\cal F}. Let q be a transcendental element over K, and let K(q) be the function field over K. We set {\cal F}_q = {\cal F} \otimes_K K(q), and to get the actions of the E_i and F_i on {\cal F}_q, we’ll use almost the same formulas as above, but we’ll need a bit more partition notation.

Given a partition, a box is removable (resp. indent) if it can be removed (resp. added) to obtain another Young diagram. Let \lambda and \mu be two partitions such that \lambda is obtained from \mu by adding an i-node \gamma to it. Let I^r_i(\lambda, \mu) (resp. R^r_i(\lambda, \mu)) be the number of indent i-nodes of \lambda (resp. number of removable i-nodes of \lambda) which are strictly to the right of \gamma. Also set N^r_i(\lambda, \mu) = I^r_i(\lambda, \mu) - R^r_i(\lambda, \mu). Define the same numbers with the superscript r replaced by l by replacing “right” with “left.” Then we set F_is_\lambda = \sum_\mu q^{N_i^r(\lambda, \mu)} s_\mu (resp. E_is_\nu = \sum_\mu q^{-N_i^l(\mu, \nu)} s_\mu) where the sum is over all \mu such that \mu / \lambda is an i-node (resp. \mu / \nu is an i-node). We also set D^{\pm}s_\lambda = q^{\pm N_0(\lambda)}s_\lambda and define K^{\pm}_is_\lambda = q^{\pm K(i, \lambda)}s_\lambda where K(i, \lambda) is the number of removable and indent i-nodes of \lambda. And we can take the quantum group U_q(\widehat{\mathfrak{sl}_n}) to be the K(q) algebra spanned by these generators.

We’ll use a family of operators V_k to find a nice basis. To define their actions, we need some definitions about ribbons. First, an m-ribbon is a connected skew Young diagram with m boxes which does not contain a 2 \times 2 square. The most northeast box of an m-ribbon is called its origin. Its spin is the number of rows it has minus 1. A connected union of m-ribbons is a horizontal m-ribbon strip if it is a skew Young diagram, and if the origin of each ribbon does not lie below another box in the same column. The weight of a horizontal m-ribbon strip is the number of m-ribbons used to build it. Any tiling of a horizontal m-ribbon strip subject to these constraints is unique, so we can define the spin of a horizontal m-ribbon strip to be the sum of the spins of its ribbons. We define V_k s_\lambda = \sum_\mu (-q)^{{\rm spin}(\mu/\lambda)} s_\mu where the sum is over all \mu such that \mu / \lambda is a horizontal n-ribbon strip of weight k. This sort of looks like the definitions one uses to define the Murnaghan–Nakayama rule for multiplying a Schur function and power sum symmetric function. In fact, in the classical limit “q=1″, V_k reduces to multiplication by the plethysm h_n \circ p_k = h_n(x_1^k, x_2^k, \dots ).

We introduce a K-linear bar involution on K(q) via q \mapsto q^{-1}, and then extend this to a compatible K-linear involution x \mapsto \overline{x} on {\cal F}_q by requiring that it commute with the actions of F_i and V_k on {\cal F}_q, and that it fixes the basis vector s_\emptyset. Let L (resp. L^-) be the free {\bf Z}[q]-submodule (resp. {\bf Z}[q^{-1}]-submodule) of {\cal F}_q spanned by the basis \{s_\lambda\}. Then we have the following theorem.

Theorem. There exist two unique bar-invariant bases B = \{G(\lambda)\} and B^- = \{G^-(\lambda)\} of {\cal F}_q such that G(\lambda) = s_\lambda \pmod {qL} and G^-(\lambda) = s_\lambda \pmod {q^{-1}L^-}.

The two bases are called the canonical basis and dual canonical basis of {\cal F}_q. They have a lot of nice properties. Going back to highest weight vectors, it turns out that G^-(n\lambda) is a highest weight vector for all \lambda. Furthermore, “setting q=1” this basis of highest weight vectors reduces to the plethysms s_\lambda \circ p_n = s_\lambda(x_1^n, x_2^n, \dots) (this is related to the classical limit of the operators V_k). Since it comes from a more “rigid” basis, we might be satisfied with this choice for a basis of highest weight vectors in {\cal F}. Another nice property which happens with canonical bases is a nonnegativity property: write G(\mu) = \sum_\lambda d_{\lambda, \mu}(q) s_\lambda and G^-(\lambda) = \sum_\mu e_{\lambda, \mu}(-q^{-1}) s_\mu where the d and e are polynomials.

Theorem.The polynomials d and e have nonnegative coefficients as polynomials in q. Furthermore, d_{\lambda, \mu}(q) is nonzero only if \lambda \le \mu and similarly, e_{\lambda, \mu}(q) is nonzero only if \mu \le \lambda.

Here we are using the dominance order on partitions: \lambda \le \mu if \lambda - \mu can be written as a nonnegative linear combination of vectors \varepsilon_i - \varepsilon_{i+1} where \varepsilon_i is the vector with a 1 in the ith coordinate and 0s in the other coordinates.

There is a bunch of other stuff which Leclerc discusses in the paper, like connections to Kazhdan-Lusztig polynomials and Macdonald polynomials, which illustrates why these canonical bases and their change of basis matrices are important, but I’ll stop here.

-Steven

]]> http://concretenonsense.wordpress.com/2009/08/31/a-fock-space-representation/feed/ 5 masnevets Tannaka–Krein duality http://concretenonsense.wordpress.com/2009/05/16/tannaka%e2%80%93krein-duality/ http://concretenonsense.wordpress.com/2009/05/16/tannaka%e2%80%93krein-duality/#comments Sat, 16 May 2009 01:17:15 +0000 Steven Sam http://concretenonsense.wordpress.com/?p=470 ]]>

In this post, I want to discuss to what extent a group’s character table determines it up to isomorphism.

First, let’s do the easy case of Abelian groups. The set of characters of an Abelian group G is itself a group called G^\vee given by pointwise multiplication. In fact, G is isomorphic to G^\vee (though not canonically) as follows: we can write G \cong {\bf Z}/q_1 \oplus \cdots \oplus {\bf Z}/q_n where the q_j are prime powers. Fix generators for each cyclic summand. For each j, the function f_j \colon G \to {\bf C} given by sending the generator of {\bf Z}/q_j to \text{exp}(2\pi i/q_j) gives an element of order q_j in G^\vee, and the map G \to G^\vee given by (a_1, \dots, a_n) \mapsto (f_1^{a_1}, \dots, f_n^{a_n}) is injective. Since the number of characters of G is equal to the order of G, we conclude that it is an isomorphism. To do this canonically (without picking the direct sum decomposition), we can just do it twice (picking the decomposition twice ends up cancelling the fact that we made a choice), and we get an isomorphism G \to (G^\vee)^\vee by sending x to the character of G^\vee defined by evaluation: \psi \mapsto \psi(x). This is a special instance of Pontryagin duality, which holds more generally for any locally compact Abelian group. So we know that the characters determine the group up to isomorphism in this case.

Now let’s look at the noncommutative case. The first value of n for which there exist two nonisomorphic noncommutative groups of order n is 8, in which case we have the dihedral group D_4 which is the symmetries of the square, and the quaternion group Q_8.

The group D_4 has the presentation \langle a,b \mid a^4 = b^2 = 1,\ b^{-1}ab = a^{-1} \rangle. For a character, we need to pick a fourth root of unity for a, along with a sign for b, and the relation b^{-1}ab = a^{-1} becomes a^2 = 1. Hence we see the four characters: assign \pm 1 to a and b in all possible ways. The other representation can be defined by sending a and b to the matrices \left[ \begin{matrix} i & 0 \\ 0 & -i \end{matrix} \right] and \left[ \begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix} \right].

The conjugacy classes are \{1\}, \{a^2\}, \{a,a^3\}, \{b,a^2b\}, \{ab, a^3b\}, so we get the following character table:
\begin{array}{ccccc} 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & -1 & -1 \\ 1 & 1 & -1 & 1 & -1 \\ 1 & 1 & -1 & -1 & 1 \\ 2 & -2 & 0 & 0 & 0 \end{array}

For the quaternions, we use the presentation Q_8 = \langle a, b \mid a^4 = 1,\ b^2 = a^2,\ b^{-1}ab = a^{-1} \rangle. Comparing with the usual definition as \{\pm 1, \pm i, \pm j, \pm k\}, we can take a=i and b=j. This presentation shows that images of (a,b) under the 4 characters must be \{(1,1), (1,-1), (-1,1), (-1,-1)\}. For the 2-dimensional representation, we can use the standard matrix representation of the quaternions: i \mapsto \left[ \begin{matrix} i & 0 \\ 0 & i \end{matrix} \right],\ j \mapsto \left[ \begin{matrix} 0 & 1 \\ -1 & 0 \end{matrix} \right],\ k \mapsto \left[ \begin{matrix} 0 & i \\ i & 0 \end{matrix} \right].

The conjugacy classes are \{1\}, \{a^2\}, \{a,a^3 \}, \{b,a^2b\}, \{ab,a^3b\}. This looks familiar, and in fact, the character table is exactly the same as that of D_4.

So there is no hope of recovering the group from its character table, but in fact, one can reconstruct the group from its representations. More precisely, we need to look at the category of its representations Rep(G). So far it still looks bad because the dimensions of the hom spaces are given by inner product of characters, and the tensor product is given by multiplication, so there must be some additional information in there somewhere. In fact there is, as Tannaka pointed out. We consider not just Rep(G), but also its forgetful functor F to Vect, the category of finite-dimensional complex vector spaces. First we note that an element x of G gives a natural transformation \pi(x) from F to itself by defining \pi(x)_V \colon F(V) \to F(V) to be multiplication by x whenever V is a representation. This association realizes G as a subgroup of the monoid of natural transformations from F to itself. Is there a way to intrinsically characterize this subgroup?

We notice two more properties of \pi(x). First, it’s the identity map on the trivial representation of G, and it preserves tensor products in the sense that \pi(x)_{V \otimes W} = \pi(x)_V \otimes \pi(x)_W as maps F(V \otimes W) \to F(V \otimes W) (we’ll call both of these properties tensor-preserving). It is also self-conjugate: Given a vector space V, we define a conjugate space \overline{V} = \{\overline{x} \mid x \in V\} by \overline{x+y} = \overline{x} + \overline{y} and a\overline{x} = \overline{a}x for a \in {\bf C}. Then if V is a representation, then so is \overline{V} by saying that g\overline{x} = \overline{g}x where \overline{g} as a matrix is g with its entries conjugated. We can conjugate a natural transformation u by defining \overline{u}_V(x) = \overline{u_{\overline{V}}(\overline{x})} for V a representation and x in V. In fact, these two properties are enough: Tannaka’s theorem says that all tensor-preserving self-conjugate natural transformations from F to itself are of the form \pi(x) for some x in G. Just as in the case of Pontryagin duality, Tannaka’s theorem holds more generally for arbitrary compact groups (one can define a topology on the set of endomorphisms of F).

So where’s the Krein and where’s the duality? Given the above information, one should think of Rep(G) as a sort of dual to G, and Krein classified the categories which are of the form Rep(G). Given a category C of vector spaces with a tensor product and an involution (the conjugation above), then C is dual to a compact group G if and only if the following three properties hold:

  1. (Identity axoim) There exists I, which is unique up to isomorphism, such that A \otimes I \cong A for all A.
  2. (Krull–Schmidt axoim) Every V can has a minimal direct sum decomposition (the summands are not isomorphic to nontrivial direct sums).
  3. (Schur’s lemma axiom) If A and B are minimal (with respect to direct sum), then Hom(A,B) is 1-dimensional if A and B are isomorphic, and is 0 otherwise.

Then C = Rep(G) where G is the tensor-preserving self-conjugate endomorphisms of the forgetful functor F.

Things don’t end here of course, there’s extensions to quantum groups and algebraic groups (Grothendieck’s Galois theory), and other things. If I ever learn this stuff, I’ll try to write a sequel to this post.

-Steven

]]> http://concretenonsense.wordpress.com/2009/05/16/tannaka%e2%80%93krein-duality/feed/ 6 masnevets The Borel–Weil–Bott theorem http://concretenonsense.wordpress.com/2009/04/15/the-borel%e2%80%93weil%e2%80%93bott-theorem/ http://concretenonsense.wordpress.com/2009/04/15/the-borel%e2%80%93weil%e2%80%93bott-theorem/#comments Wed, 15 Apr 2009 01:21:43 +0000 Steven Sam http://concretenonsense.wordpress.com/?p=395 ]]>

In connection with my last post on the Boij–Söderberg conjectures, I mentioned constructing equivariant supernatural vector bundles and equivariant pure Cohen–Macaulay modules using the Borel–Weil–Bott theorem. So in this post, I’d like to say something about what this theorem says and next time discuss how it can be used. I learned the stuff on Bott’s theorem from Jerzy Weyman’s book Cohomology of Vector Bundles and Syzygies [warning: there are some mistakes in the statement of Bott's theorem for general reductive groups]. Bott’s theorem is usually stated for a reductive group, but for concreteness we’ll stick with the general linear group, since that’s all we’ll need.

The setup is as follows: k denotes some field of characteristic 0, G = \mathbf{GL}(n), B is the subgroup of upper triangular matrices, and T is the subgroup of diagonal matrices. Then G/B is the complete flag variety whose k-points correspond to maximal flags V_\bullet = (0 \subset V_1 \subset \cdots \subset V_n = k^n) where \dim V_i = i. We are interested in realizing representations of G as cohomology groups of line bundles over G/B. But first we’ll state the relative version of the theorem.

For this, let X be any variety (really, X could be any scheme) and \mathcal{E} be a vector bundle of rank n over X. We can construct the relative flag variety as follows. For any affine open set U inside of X where we get a trivialization \mathcal{E}|_U = U \times k^n, let \mathrm{Fl}(\mathcal{E},U) be the complete flag variety of k^n. Using the gluing data of \mathcal{E}, we can also glue together the spaces \mathrm{Fl}(\mathcal{E}, U) \times U (on intersections, we have an isomorphism k^n \to k^n, and this defines an isomorphism of complete flag varieties) to get a scheme \mathrm{Fl}(\mathcal{E}) which maps to X, call the map h. In fact, h is a locally trivial fibration whose fibers are complete flag varieties. To define the analogues of R_i, consider the vector bundle h^*(\mathcal{E}) = \{(x, F_\bullet, y) \mid x \in X,\ F_\bullet \in \mathrm{Fl}(\mathcal{E})|_x,\ y \in \mathcal{E}|_x\} on \mathrm{Fl}(\mathcal{E}). We define \mathcal{R}_i to be the subbundle of h^*(E) consisting of points where y \in F_i. Note that if we took X to be a single point (in particular Spec(k)), then \mathcal{E} is just a k-vector space, and \mathrm{Fl}(\mathcal{E}) is the usual flag variety. So we can also define, in analogy, \mathcal{L}(\alpha) = \mathcal{R}_1^{-\alpha_1} \otimes (\mathcal{R}_2 / \mathcal{R}_1)^{-\alpha_2} \otimes \cdots \otimes (\mathcal{E} / \mathcal{R}_n)^{-\alpha_n}.

Now let W be the Weyl group of G, i.e., the normalizer of T in G quotiented by T, in our case W is the symmetric group on n letters. In fact, W is a finite Coxeter group, and hence is equipped with a length function, denoted \ell. For a permutation w, \ell(w) is just the minimal number of simple transpositions i \leftrightarrow i+1 you need when expressing w as a product of them. So W has a natural action on the set of weights of T, and we define a dotted action of W on the weight lattice as follows. First, let 2\rho be the sum of the positive roots (i.e., the weights that appear when T acts on B via conjugation). In our case, we have \rho = (n-1, n-2, \dots, 1, 0). Then for w \in W and \alpha a weight, we set w^\bullet(\alpha) = w(\alpha + \rho) - \rho. Let K_\beta denote the Weyl functor, or co-Schur functor associated to the weight \beta. I’ll describe its construction in a later post. For now, it’s a functorial way of constructing, from E, an irreducible representation of \mathbf{GL}(E) of highest weight \beta. We have the following theorem:

Theorem. With the notation above, one of two mutually exclusive cases occurs:

  1. There exists a nonidentity element w \in S_n such that w^\bullet(\alpha) = \alpha. In this case, all higher direct images \mathrm{R}^ih_*(\mathcal{L}(\alpha)) vanish.
  2. Otherwise, there is a unique w \in S_n such that \beta = w^\bullet(\alpha) is a dominant weight. In this case, if i \ne \ell(w), then \mathrm{R}^ih_*^i(\mathcal{L}(\alpha)) = 0, and \mathrm{R}^{\ell(w)}h_*(\mathcal{L}(\alpha)) = K_\beta(\mathcal{E}^*) is an equivariant bundle whose fibers are the dual of the irreducible representation of \mathbf{GL}(n) with highest weight \beta.

This gives us the cohomology for line bundles over G/B since when X is a point (and hence \mathcal{E} is just a vector space), the higher direct images become cohomology groups.

If we assume this theorem, we can use the Leray spectral sequence to extract some information about vector bundles on partial flag varieties. Let b = (0=b_0, b_1, \dots, b_r) be a sequence of increasing numbers, and let \mathrm{Fl}(b; \mathcal{E}) be the relative partial flag variety (i.e., the fibers are flags F_\bullet = (F_{i_1} \subset F_{i_2} \subset \cdots \subset F_{i_r} \subset k^n) where \dim F_{i_j} = b_j. So we still have the tautological bundles \mathcal{R}_{b_i}. For each r>j>0, let \alpha^j be a nonincreasing sequence of numbers of length b_j - b_{j-1}, and write \alpha = (\alpha^1, \dots, \alpha^r) (this is an n-tuple of numbers, not an r-tuple of sequences). Then we can define a bundle \mathcal{V}(\alpha) = K_{\alpha^1}(\mathcal{R}_{b_1})^* \otimes \cdots \otimes K_{\alpha^r}(\mathcal{R}_{b_r} / \mathcal{R}_{b_{r-1}})^*. Now Bott’s theorem is the same as above except \mathrm{Fl}(b; \mathcal{E}) and \mathcal{V}(\alpha) replace \mathrm{Fl}(\mathcal{E}) and \mathcal{L}(\alpha), respectively.

To deduce this version, note that we have a forgetful morphism f \colon \mathrm{Fl}(\mathcal{E} \to \mathrm{Fl}(b; \mathcal{E}), which is a locally trivial fibration whose fibers are products of complete flag varieties. On \mathrm{Fl}(\mathcal{E}), we can construct the line bundle \mathcal{L}(\alpha) as before. Using the relative version of Bott’s theorem and induction on the number of terms that we forget, we see that f_*(\mathcal{L}(\alpha)) = \mathcal{V}(\alpha), and its higher direct images vanish because we assumed that each \alpha^j was a dominant weight. Letting h \colon \mathrm{Fl}(b; \mathcal{E}) \to X be the structure map, we can also use the relative Bott’s theorem to calculate \mathrm{R}^i(hf)_*(\mathcal{L}(\alpha)). Alternatively, we have the Leray spectral sequence \mathrm{E}_2^{p,q} = (\mathrm{R}^ph_* \circ \mathrm{R}^qf_*)(\mathcal{L}(\alpha)) \Rightarrow \mathrm{R}^{p+q}(hf)_*(\mathcal{L}(\alpha)). But we just showed that these terms are 0 for q=0, so the spectral sequence degenerates and we conclude that \mathrm{R}^ph_*(\mathcal{V}(\alpha)) = \mathcal{R}^p(hf)_*(\mathcal{L}(\alpha)).

I’ll end this post with what this all says on projective space, and next time I’ll discuss how we can use this special case to construct equivariant pure modules and supernatural vector bundles.

Over projective space \mathbf{P}(E) \cong \mathbf{P}^{n-1}_k, the fiber of the tautological bundle \mathcal{R} over a point (which is a line in E \cong k^n) is the simply the vectors lying in that line. This is also the total space of the dual of the Serre twisting sheaf \mathcal{O}_{\mathbf{P}(E)}(-1). Given the inclusion \mathcal{R} \subset E \times \mathbf{P}(E), the quotient will be called \mathcal{Q} and called the tautological quotient. Then our bundles of interest are of the form K_u\mathcal{R} \otimes K_\alpha \mathcal{Q} = (K_\alpha \mathcal{Q})(-u) where u is an integer and \alpha = (\alpha_1 \ge \alpha_2 \ge \cdots \ge \alpha_{n-1}). Note that this is the same \mathcal{V}(\gamma) where \gamma = (-u, -\alpha_{n-1}, ..., -\alpha_1) by duality. Hence the theorem says

Theorem. With the notation above, one of two mutually exclusive cases occurs:

  1. There exists a nonidentity element w \in S_n such that w^\bullet(\alpha_1, \dots, \alpha_{n-1}, u) = \alpha. In this case, all cohomology groups \mathrm{H}^i(\mathbf{P}(E); K_\alpha\mathcal{Q}(-u)) vanish.
  2. Otherwise, there is a unique w \in S_n such that \beta = w^\bullet(\alpha_1, \dots, \alpha_{n-1}, u) is nondecreasing. In this case, if i \ne \ell(w), then \mathrm{H}^i(\mathbf{P}(E); (K_\alpha \mathcal{Q})(-u)) = 0, and \mathrm{H}^{\ell(w)}(\mathbf{P}(E); (K_\alpha \mathcal{Q})(-u)) = K_\beta(E) is an irreducible representation of \mathbf{GL}(E) with highest weight \beta.

-Steven

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