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.



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:


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)).



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:



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. ♦



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.


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).


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


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}.


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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