Polynomials and Representations XXXVII

Notations and Recollections

For a partition \lambda\vdash d, one takes its Young diagram comprising of boxes. A filling is given by a function T:\lambda \to [m] for some positive integer m. When m=d, we will require the filling to be bijective, i.e. T contains {1,…,d} and each element occurs exactly once.

If w\in S_m and T:\lambda \to [m] is a filling, then w(T) = w\circ T is obtained by replacing each i in the filling with w(i). For a filling T, the corresponding row (resp. column) tabloid is denoted by {T} (resp. [T]).

Recall from an earlier discussion that we can express the S_d-irrep V_\lambda as a quotient of \mathbb{C}[S_d]b_{T_0} from the surjection:

\mathbb{C}[S_d] b_{T_0} \to \mathbb{C}[S_d] b_{T_0} a_{T_0}, \quad v \mapsto v a_{T_0}.

Here T_0 is any fixed bijective filling \lambda \to [d].

Concretely, a C-basis for \mathbb{C}[S_d]b_{T_0} is given by column tabloids [T] and the quotient is given by relations: [T] = \sum_{T'} [T'] where T’ runs through all column tabloids obtained from T as follows:

  • fix columns jj’ and a set B of k boxes in column j’ of T; then T’ is obtained by switching B with a set of k boxes in column j of T, while preserving the order. E.g.

modulo_relations_for_specht

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For Representations of GLn

From the previous article we have V(\lambda) = V^{\otimes d} \otimes_{\mathbb{C}[S_d]} V_\lambda, where V_\lambda is the quotient of the space of column tabloids described above. We let V^{\times \lambda} be the set of all functions \lambda \to V, i.e. the set of all fillings of λ with elements of V. We define the map:

\Psi : V^{\times\lambda} \to V^{\otimes d}\otimes_{\mathbb{C}[S_d]} V_\lambda, \quad (v_s)_{s\in\lambda} \mapsto \overbrace{\left[v_{T^{-1}(1)} \otimes \ldots \otimes v_{T^{-1}(d)}\right]}^{\in V^{\otimes d}} \otimes [T]

for any bijective filling T:\lambda \to [d]. This is independent of the T we pick; indeed if we replace T by w(T) = w\circ T  for w\in S_d, the resulting RHS would be:

\begin{aligned}\left[v_{T^{-1}w^{-1}(1)} \otimes \ldots \otimes v_{T^{-1}w^{-1}(d)}\right] \otimes [w(T)] &= \left[v_{T^{-1}w^{-1}(1)}\otimes \ldots \otimes v_{T^{-1} w^{-1}(d)}\right]w \otimes [T]\\ &= \left[v_{T^{-1}(1)} \otimes \ldots \otimes v_{T^{-1}(d)}\right] \otimes [T]\end{aligned}

where the first equality holds since the outer tensor product is over \mathbb{C}[S_d] and the second equality follows from our definition (v_1' \otimes \ldots \otimes v_d')w = v_{w(1)}' \otimes \ldots \otimes v_{w(d)}'. Hence \Psi is well-defined. It satisfies the following three properties.

Property C1. \Psi is multilinear in each component V.

In other words, if we fix s\in \lambda and consider \Psi as a function on V in component s of V^{\times\lambda}, then the resulting map is C-linear. E.g. if w'' = 2w + 3w', then:

c1_multilinear

This is clear.

Property C2. Suppose (v_s), (v'_s)\in V^{\times\lambda} are identical except v'_s = v_t and v'_t = v_s, where s,t\in \lambda are in the same column. Then \Psi((v'_s)) = -\Psi((v_s)).

c2_alternating

Proof

Let w\in S_d be the transposition swapping s and t. Then w([T]) = -[T] by alternating property of the column tabloid and w^2 = e. Thus:

\begin{aligned}\left[v'_{T^{-1}(1)} \otimes \ldots \otimes v'_{T^{-1}(d)}\right] \otimes [T] &= \left[ v_{T^{-1}w^{-1}(1)} \otimes \ldots \otimes v_{T^{-1}w^{-1}(d)}\right] \otimes -w([T])\\ &= -\left[v_{T^{-1}(1)} \otimes\ldots \otimes v_{T^{-1}(d)}\right]\otimes [T]. \end{aligned} ♦

Finally, we have:

Property C3. Let (v_s)\in V^{\times\lambda}. Fix two columns j<j' in the Young diagram for λ, and a set B of k boxes in column j’. As A runs through all sets  of k boxes in column j, let (v_s^A) \in V^{\times\lambda} be obtained by swapping entries in A with entries in B while preserving the order. Then:

\displaystyle \Psi((v_s)) = \sum_{|A| = |B|} \Psi((v_s^A)).

E.g. for any u,v,w,x,y,z\in V we have:

c3_column_swaps

Proof

Fix a bijective filling T:\lambda \to [d]. Then:

\begin{aligned}\Psi((v_s^A)) &= \left[v_{T^{-1}(1)}^A \otimes \ldots \otimes v_{T^{-1}(d)}^A\right] \otimes [T ]\\ &= \left[v_{T^{-1}w^{-1}(1)} \otimes \ldots \otimes v_{T^{-1}w^{-1}(d)}\right] \otimes [T] \\ &= \left[v_{T^{-1}(1)} \otimes \ldots \otimes v_{T^{-1}(d)}\right] \otimes w([T])\end{aligned}

where w\in S_d swaps the entries in A with those in B while preserving the order (note that w^2 =e). But the sum of all such w([T]) vanishes in V_\lambda. Hence \sum_A \Psi((v_s^A)) = 0. ♦

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Universality

Definition. Let V, W be complex vector spaces. A map \Psi : V^{\times \lambda} \to W is said to be λ-alternating if properties C1, C2 and C3 hold.

The universal λ-alternating space (or the Schur module) for V is a pair (F(V), \Phi_V) where

  • F(V) is a complex vector space;
  • \Phi_V : V^{\times\lambda} \to F(V) is a λ-alternating map,

satisfying the following universal property: for any λ-alternating map \Psi : V^{\times\lambda} \to W to a complex vector space W, there is a unique linear map \alpha : F(V) \to W such that \alpha\circ \Phi_V = \Psi.

F(V) is not hard to construct: the universal space which satisfies C1 and C2 is the alternating space:

\displaystyle \left(\text{Alt}^{\mu_1} V\right) \otimes \ldots \otimes \left(\text{Alt}^{\mu_e}V\right), \quad \mu := \overline\lambda.

So the desired F(V) is obtained by taking the quotient of this space with all relations obtained by swapping a fixed set B of coordinates in \text{Alt}^{j'} with a set A of coordinates in \text{Alt}^j, and letting A vary over all |A| = |B|. E.g. the relation corresponding to our above example for C3 is:

\begin{aligned} &\left[ (u\wedge x\wedge z) \otimes (v\wedge y) \otimes w\right] -\left[ (u\wedge y\wedge z) \otimes (u\wedge x) \otimes w\right] \\ - &\left[ (v\wedge x\wedge y)\otimes (u\wedge z)\otimes w\right] - \left[ (u\wedge x\wedge w) \otimes (v\wedge z) \otimes w\right]\end{aligned}

over all u,v,w,x,y,z\in V.

By universality, the λ-alternating map \Psi: V^{\times\lambda} \to V^{\otimes d} \otimes_{\mathbb{C}[S_d]} V_\lambda thus induces a linear:

\alpha: F(V) \longrightarrow V^{\otimes d} \otimes_{\mathbb{C}[S_d]} V_\lambda.

You can probably guess what’s coming next.

Main Theorem. The above \alpha is an isomorphism.

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Proof of Main Theorem

First observe that \alpha is surjective by the explicit construction of F(V) so it remains to show injectivity via dim(LHS) ≤ dim(RHS).

Now V^{\otimes d}\otimes_{\mathbb{C}[S_d]}V_\lambda \cong V(\lambda), and we saw earlier that its dimension is the number of SSYT with shape λ and entries in [n].

On the other hand, let e_1, \ldots, e_n be the standard basis of V= \mathbb{C}^n. If T is any filling with shape λ and entries in [n], we let e_T be the element of F(V) obtained by replacing each i in T by e_i \in V; then running through the map \Phi_V: V^{\times \lambda} \to F(V).

basis_element_of_schur_module

Claim. The set of e_T generates F(V), where T runs through all SSYT with shape λ and entries in [n].

Proof

Note that the set of e_T, as T runs through all fillings with shape λ and entries in [n], generates F(V).

Let us order the set of all fillings of T as follows: T’ > T if, in the rightmost column j where T’ and T differ, at the lowest (i,j) in which T_{ij}' \ne T_{ij}, we have T_{ij}' > T_{ij}.

comparison_of_two_fillings

This gives a total ordering on the set of fillings. We claim that if T is a filling which is not an SSYT, then e_T is a linear combination of e_S for S > T.

  • If two entries in a column of T are equal, then e_T = 0 by definition.
  • If a column j and row i of T satisfy T_{i,j} > T_{i+1,j}, assume j is the rightmost column for which this happens, and in that column, i is as large as possible. Swapping entries (i,j) and (i+1, j) of T gives us T’T and e_T = -e_{T'}.
  • Now suppose all the columns are strictly ascending. Assume we have T_{i,j} > T_{i, j+1}, where j is the largest for which this happens, and T_{k,j} \le T_{k,j+1}, for k=1,\ldots, i-1. Swapping the topmost i entries of column j+1, with various  i entries of column j, all the resulting fillings are strictly greater than T. Hence e_T = -\sum_S e_S, where each S > T.

Thus, if T is not an SSYT we can replace e_T with a linear combination of e_S where S > T. Since there are finitely many fillings T (with entries in [n]), this process must eventually terminate so each e_T can be written as a linear sum of e_S for SSYT S. ♦

Thus \dim F(V) ≤ number of SSYT with shape λ and entries in [n], and the proof for the main theorem is complete. From our proof, we have also obtained:

Lemma. The set of \{e_T\} forms a basis for F(V), where T runs through the set of all SSYT with shape λ and entries in [n].

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