A quadratic identity in the shuffle algebra and an alternative proof for de Bruijn's formula
aa r X i v : . [ m a t h . R A ] J un A quadratic identity in the shuffle algebra andan alternative proof for de Bruijn’s formula
Laura Colmenarejo ∗ Joscha Diehl † Miruna-Ştefana Sorea ‡ June 19, 2020
Abstract
In this paper we prove an intriguing identity in the shuffle algebra,first observed in [CGM19] in the setting of lattice paths. It has a closeconnection to de Bruijn’s formula when interpreted in the frameworkof signatures of paths.
Contents SL invariants and signed volume . . . . . . . . . . . . . . . . 7 ∗ L. Colmenarejo, University of Massachusetts at Amherst (USA) † J. Diehl, University of Greifswald (Germany) ‡ M.-Ş. Sorea, Max Planck Institute for Mathematics in the Sciences Leipzig (Germany) Introduction A path is a continuous map X : r , s Ñ R d . We shall assume that thefunctions X p i q , i “ , . . . , d , are (piecewise) continuously differentiable func-tions. Describing phenomena parametrized by time, they appear in manybranches of science such as mathematics, physics, medicine or finances. Formathematics in particular, see [CGM19] for references.One way to look at paths is through their iterated-integrals ż dX p w q t . . . dX p w n q t n : “ ż ă t ă .. ă t n ă X p w q t . . . X p w n q t n dt . . . dt n , where n ě and w , . . . , w n P t , . . . , d u . The first systematic study ofthese integrals was undertaken by Kuo Tsai Chen [Che57]. For example,Chen proved that, up to an equivalence relation, iterated-integrals uniquelydetermine a path.In the field of stochastic analysis paths are usually not differentiable . Nonethe-less (stochastic) integrals have played a major role there and this culminatedin Terry Lyons’ theory of rough paths [LQ02, FH14]. Owing to their de-scriptive power of nonlinear phenomena (compare Chen’s uniqueness resultabove), these objects have recently been successfully applied to statisticallearning [LNO14, KSH `
17, KO19, LZL ` Theorem 1.1 ([CGM19]) . det ¨˚˚˚˚˝ż dX p q dX p q . . . ż dX p q dX p d q ... . . . ... ż dX p d q dX p q . . . ż dX p d q dX p d q ˛‹‹‹‹‚ “ d ˜ ÿ σ P S d sign p σ q ż dX σ p q . . . dX σ p d q ¸ . (1)Their proof is based on calculations with lattice paths. One consequenceof this result is that it gives an inequality for the iterated-integrals of ordertwo (i.e. this particular determinant is non-negative). Notice that when2orking on the Zariski closure of the space of signatures, as in the frameworkof [AFS19], one can only obtain equalities . See [CGM19, p. 22-23] for moredetails.In the current paper we look at this statement from a purely algebraic per-spective. (We advise the reader to peak ahead to Section 2 for notation usedin the following.) Consider the space of formal linear combinations of wordsin the alphabet r d s “ t , . . . , d u , and denote it by T p R d q . Given two words w and v in the alphabet r d s , the shuffle product of them, w (cid:1) v , is definedas the sum of all possible shuffle products that keep the respective order ofthe two words intact. For example, (cid:1) “ ` ` ` ` ` Then, ` T p R d q , (cid:1) ˘ is an algebra known as the shuffle algebra . In a sense madeprecise in Section 2 the determinant (in the shuffle algebra) of the matrix ¨˚˝ . . . ... . . . ... d1 . . . dd ˛‹‚ relates to the determinant (in R ) appearing in (1). To reformulate the otherside of the equality in (1), we recall the basis of SL p R d q -invariants in T p R d q ,going back to Weyl [Wey46], and presented, in the language relevant usedhere, in [DR19]. This basis is indexed by standard Young tableaux of shape p w, . . . , w q looooomooooon d times , for w ě . For us, only the cases w “ , are relevant and wedenote the basis by t inv p T qu , where T ranges over these standard Youngtableaux. The definition of this basis is presented in Section 2.4, and weillustrate with two examples here: inv ˜ ¸ “ ´ , and inv ˜ ¸ “ ´ ´ ` . (2)We can now state our main theorem (see Theorem 4.1 in Section 4). Theorem.
For d ě the following two equalities hold: det (cid:1) ¨˚˝ . . . ... . . . ... d1 . . . dd ˛‹‚ “ inv ¨˚˚˚˚˝ ... ... d ´ d ˛‹‹‹‹‚ “ d inv ¨˚˚˚˚˝ ... d ˛‹‹‹‹‚ (cid:1) .
3n particular, we give a new and purely algebraic proof of the fact that thedeterminant of the second level of the iterated-integrals signature is a square,Theorem 1.1 (see Corollary 4.2). As an application of our approach, weobtain the de Bruijn’s formula for the Pfaffian. We note that de Bruijn’s for-mula was looked at in the language of shuffle algebras already in [LT02]. Ourviewpoint differs in that we approach the topic through invariant theory.In Section 2, we present the framework of the shuffle algebra and the half-shuffle products, together with their relation with the signatures of paths.Since our proof for the main theorem uses some results from representationtheory, we include in Section 2.3 the necessary basic notions. Moreover,in Section 2.4 we relate our setting with the work of [DR19] about SL n -invariants and signed volume. In Section 3 we present already known resultsthat will help us to prove our main results. For instance, we present severalidentities related to the shuffle product that already appeared in [And83].Section 4 is dedicated to prove our main theorem. In Section 5, we presentthe relation of our main theorem with the de Bruijn’s formula for both theeven and odd-dimensional cases. Acknowledgements
The authors are grateful to Bernd Sturmfels and Mateusz Michałek for help-ful discussions and suggestions, and for bringing the team together. Theauthors are also grateful to the MPI MiS for giving us the opportunity towork on this project in such a nice environment. We would like to thankDarij Grinberg for making us aware of several inconsistencies in an earlierversion.
In this section we present several connected frameworks. First, we talk aboutthe shuffle algebra and the half-shuffle operation. A survey for the historyof the shuffle product can be found in [FP13] and our presentation here isinspired by [CP18], especially the part concerning the half-shuffle operation.Next, we describe briefly the relation between the setting of signatures ofpaths and the shuffle algebra. We refer the reader to [DR19, Subsection2.1] for more details. We also include a brief summary of the representationtheory setting, which can be found in [FH91] with full detail. We finish bypresenting a brief overview of the SL p R d q -invariants inside of T p R d q .4 .1 The shuffle algebra and half–shuffle products Let us consider the alphabet r d s “ t , . . . , d u , together with the empty word ε , and denote by T pp R d qq the space of infinite formal linear combinations of words in the alphabet. Together with the concatenation product, w ¨ v , ` T pp R d qq , ¨ ˘ is an algebra, known as the (completed) tensor algebra .Its algebraic dual, denoted by T p R d q , consists of finite linear combinations ofwords in r d s . We consider the algebra ` T p R d q , (cid:1) ˘ , also known as the shufflealgebra , with the operation (cid:1) defined as follows. Definition 2.1.
Let us consider the words u , v and w , and the letters a and b . The shuffle product of two words is defined recursively by ε (cid:1) u “ u (cid:1) ε “ u , and p v ¨ a q (cid:1) p w ¨ b q “ p v (cid:1) p w ¨ b qq ¨ a ` pp v ¨ a q (cid:1) w q ¨ b . It is extended bilinearily to a commutative product on all of T p R d q .The shuffle product can be seen as the symmetrisation of the right half-shuffleproduct. Definition 2.2. [FP13, Definition 1],[EML53, Section 18] The right half-shuffle product is recursively given on words as w ą i : “ wi , and w ą vi : “ p w ą v ` v ą w q ¨ i , where w , v are words and i is a letter.Note that if w , v are any non-empty words, then w (cid:1) v “ w ą v ` v ą w .Hence in Definition 2.2 we can replace w ą vi “ p w ą v ` v ą w q ¨ i withthe equivalent equality w ą vi “ p w (cid:1) v q ¨ i . In general, for non-emptywords, one has u ą p v ą w q “ p u (cid:1) v q ą w . (3)In particular, notice that the shuffle product is associative, while the half-shuffle product is not. For a (piecewise) smooth path X : r , s Ñ R d the collection of iteratedintegrals ż dX p w q t . . . dX p w n q t n : “ ż ă t ă .. ă t n ă X p w q t . . . X p w n q t n dt . . . dt n , n ě and w , . . . , w n P r d s is conveniently stored in the iterated-integrals signature σ p X q : “ ÿ n ě , w “ w ... w n Pr d s ż dX p w q t . . . dX p w n q t n dt . . . dt n w . This object is a formal infinite sum of words w in the alphabet r d s whosecoefficients are given by the integrals. That is, σ p X q is an element of the(completed) tensor algebra T pp R d qq .Alternatively, σ p X q can be seen as a linear function on T p R d q given by A w , σ p X q E “ ż dX p w q t . . . dX p w n q t n , for any element w P r d s , and extended linearly to all of T p R d q . It turns outthat σ p X q is in fact a multiplicative character (i.e. an algebra morphisminto R ) on p T p R d q , (cid:1) q , see [Reu93, Corollary 3.5]. This fact is also calledthe shuffle identity and reads as A w (cid:1) v , σ p X q E “ A w , σ p X q E ¨ A v , σ p X q E , @ w , v P T p R d q . (4) This section provides some broad ideas from representation theory that areused in this paper. For more details, we refer the reader to [FH91].Given a group G and an n -dimensional vector space over R , V , we say thatthe map ρ : G ÝÑ GL p V qq is a representation of G if ρ is a group homomor-phism, where GL p V q is the group of automorphism of V . In general, ρ isidentified with V , and ρ p g qp v q : “ g ¨ v , for all v P V and g P G . In this sense,we say that the group acts (on the left) on the vector space. There aredifferent ways to construct representations from other representations; forinstance, by constructing the direct sum or the tensor product of represen-tations, or restricting or inducing representation from groups to subgroups,and vice versa.Another approach to understand the representations is by looking at themas modules, which allows us to study representations as vector spaces overthe group algebra R r G s (i.e. set of all linear combinations of elements in G with coefficients in R ).A (non-zero) representation V of G is said to be irreducible if the only sub-spaces of V invariant under the action of G are the vector space itself and thetrivial subspace. By Schur’s lemma and Maschke’s Theorem, we know thatgiven any representation of a well-behaved group G we can decompose it as G finite or G “ SL p R d q will do. V – À k W ‘ n k k , where the W k are pairwise non-isomorphicirreducible representations and n k denotes its multiplicity. This is called the canonical decomposition of V or the decomposition into isotypic components .The idea for isotypic components is that there may be several irreduciblerepresentations that are isomorphic. We may need only to take a “represen-tative” and count how many irreducible representation are isomorphic to it.One of the main general problems in Representation Theory is to charac-terise the irreducible representations of a group and to give an algorithm tocompute the canonical decomposition of any other representation.Each representation is also characterised by the trace of the matrix associatedto each element of the group. This vector is known as the character of therepresentation, and it is invariant under the conjugacy classes. Charactertheory is closely related to the theory of symmetric functions.In this section we focus our attention on the symmetric group and on thegeneral linear group. Let S d be the group of permutations of t , , . . . , d u , for d ě . In the case of the symmetric group S d , the conjugacy classes are inbijection with the partitions of d , which are weakly decreasing sequences ofpositive integers that sum up to d . This bijection is given by the decompo-sition of the permutations in cycles. Given a partition λ “ p λ , λ , . . . , λ ℓ q ,we associate to λ a Young diagram, which is just an array of boxes with λ i boxes in the i th row. Then, the standard Young tableaux are fillings of Youngdiagrams with all the numbers in the set t , , . . . , d u that are increasing incolumns and rows. SL invariants and signed volume The natural representation of SL p R d q on R d induces a representation on T p R d q . The study of invariants to this action goes back over a century,see [Wey46] for a good starting point of the literature. We recall the presen-tation used in [DR19].As mentioned in the introduction, a basis for the invariants is indexed by standard Young tableaux of shape p w, . . . , w q looooomooooon d times , for w ě arbitrary. Givena Young tableaux T , the corresponding invariant basis element, denoted inv p T q is obtained as follows: let n “ wd be the length of T and considerthe word w “ j j . . . j n , where j ℓ “ i if and only if ℓ is in the i th row of T . In [DR19] it is denoted by ι p e T q . T “ gives the word . Then inv p T q : “ ÿ σ sign p σ q σ w , where the sum is over all permutations σ P S n that leave the values in eachcolumn of T unchanged. inv ˜ ¸ “ p id ´p q ´ p q ` p qp qq “ ´ ´ ` . Our study only look at the following two particular standard Young tableauxfor w “ , t ,d : “ ... d , t ,d : “ ... ... d ´ d . (5)We also recall the matrix introduced before: W d : “ ¨˚˝ . . . ... . . . ... d1 . . . dd ˛‹‚ , (6)which is an element of A d ˆ d where A “ T p R d q , seen as a commutative algebra(over R ). Notation 3.1.
Please note that we use the convention of strict left bracket-ings for the half-shuffle product. For words w , . . . , w k , w ą w ą . . . ą w k : “ pp w ą w q ą . . . q ą w k . The basis element inv p t ,d q has a nice geometric interpretation in the setting ofiterated-integrals, see [DR19] where it is denoted by inv d . Recall that the half-shuffle product is non-associative, so the specification of a brack-eting is necessary. k words in terms ofhalf-shuffle products. Lemma 3.2.
For any non-empty words w , . . . , w k , the following equalityholds: w (cid:1) ¨ ¨ ¨ (cid:1) w k “ ÿ σ P S k w σ p q ą w σ p q ą . . . ą w σ p k q . (7) Proof.
We proceed by induction. For k “ , relation (7) is trivially true. Letus suppose that the equality holds for k and prove it for k ` . We have: w (cid:1) ¨ ¨ ¨ (cid:1) w k ` “ ˆ w (cid:1) ¨ ¨ ¨ (cid:1) w k ˙ (cid:1) w k ` “ ˆ ÿ σ P S k w σ p q ą . . . ą w σ p k q ˙ (cid:1) w k ` “ ˆ ÿ σ P S k w σ p q ą . . . ą w σ p k q ˙ ą w k ` ` w k ` ą ˆ ÿ σ P S k w σ p q ą . . . ą w σ p k q ˙ “ ÿ σ P S k w σ p q ą . . . ą w σ p k q ą w k ` ` ÿ σ P S k ˆ w k ` (cid:1) ˆ w σ p q ą . . . ą w σ p k ´ q ˙˙ ą w σ p k q “ ÿ σ P S k w σ p q ą . . . ą w σ p k q ą w k ` ` ˆ w k ` ą ˆ ÿ σ P S k w σ p q ą . . . ą w σ p k ´ q ˙ ` ÿ σ P S k ˆ w σ p q ą . . . ą w σ p k ´ q ˙ ą w k ` ˙ ą w σ p k q “ ÿ σ P S k w σ p q ą . . . ą w σ p k q ą w k ` ` ÿ σ P S k w σ p q ą . . . ą w σ p k ´ q ą w k ` ą w σ p k q ` ÿ σ P S k ˆ w k ` ą ˆ w σ p q ą . . . ą w σ p k ´ q ˙˙ ą w σ p k q “ . . . “ ÿ σ P S k w σ p q ą . . . ą w σ p k q ą w k ` ` ÿ σ P S k w σ p q ą . . . ą w σ p k ´ q ą w k ` ą w σ p k q ` ¨ ¨ ¨ ` ÿ σ P S k w k ` ą w σ p q ą . . . ą w σ p k ´ q ą w σ p k q “ ÿ τ P S k ` w τ p q ą . . . ą w τ p k ´ q ą w τ p k q ą w τ p k ` q . The following statement seems to first appear in [And83] (see also [dB55,p.1]) and is also known as the continuous (or generalized) Cauchy-Binetformula [Joh05, Proposition 2.10] 9 emma 3.3 ([And83]) . Let t , . . . , d u and t , . . . , d u be two alphabets in d letters each. Then det (cid:1) ¨˚˚˝
11 12 . . . . . . . . . . . . . . . . . . d1 d2 . . . dd ˛‹‹‚ “ ÿ σ,τ P S d sign p σ q sign p τ q r σ p q τ p q ą σ p q τ p q ą . . . ą σ p d q τ p d qs Proof.
For a fixed σ P S d , applying Lemma 3.2 with w i : “ i σ p i q , we get (cid:1) di “ i σ p i q “ ÿ τ P S d τ p q σ p τ ` qq ą τ p q σ p τ ` qq ą . . . ą τ p d q σ p τ p d qq . Then the left-hand side becomes: ÿ σ P S d ÿ τ P S d sign p σ q τ p q σ p τ ` qq ą τ p q σ p τ ` qq ą . . . ą τ p d q σ p τ p d qq“ ÿ σ P S d ÿ τ P S d sign p σ ˝ τ q sign p τ q τ p q σ p τ ` qq ą τ p q σ p τ ` qq ą . . . ą τ p d q σ p τ p d qq“ ÿ ρ P S d ÿ τ P S d sign p ρ q sign p τ q τ p q ρ ` q ą τ p q ρ ` q ą . . . ą τ p d q ρ p d q , as desired. In this section we present a technical lemma that is crucial for the proof ofour main result. This lemma states that, for the determinant det (cid:1) p W d q , theshuffle of letters can be replaced by a “shuffle of blocks of 2 letters”. First,let us see one example. Example 3.4.
For d “ , we have that det (cid:1) p W q “ det (cid:1) ˆ
11 1221 22 ˙ “ ` ´ ´ “ shuffle of the block with the block ´ shuffle of the block with the block . Further note, that this is equal to inv ¨˝ ˛‚ . Proposition 3.5. det (cid:1) p W d q “ ÿ σ P S d sgn p σ q ˆ ÿ τ P S d d ź i “ τ p i q σ p τ ` i q ˘˙ “ inv ¨˚˚˚˚˝ ... ... d ´ d ˛‹‹‹‹‚ , where the product in the middle is the concatenation product. Before presenting the proof for Proposition 3.5, we illustrate its idea in thecase d “ . Example 3.6.
Consider d “ . By Lemma 3.3, we obtain ÿ σ P S sgn p σ q (cid:1) i “ i σ p i q “ ÿ σ,τ P S sign p σ q sign p τ q r σ p q τ p q ą σ p q τ p q ą σ p q τ p qs“ p ą q ą ` p ą q ą ` p ą q ą ` p ą q ą ` p ą q ą ` p ą q ą ` ¨ ¨ ¨ ´ p ą q ą ´ p ą q ą ´ p ą q ą ´ p ą q ą ´ p ą q ą ´ p ą q ą “ : ‹ . Now observe that the term p ą q ą “ p ą q ¨ ` p (cid:1) (cid:1) q ¨ cancels with the term p ą q ą “ p ą q ¨ ` p (cid:1) (cid:1) q ¨ , to give the term p ą q ¨ ´ p ą q ¨ .Applying this to all terms, we replace the right-most half-shuffle product bya concatenation product. That is, ‹ “p ą q ¨ ` p ą q ¨ ` p ą q ¨ ` p ą q ¨ ` p ą q ¨ ` p ą q ¨ ` . . . ´ p ą q ¨ ´ p ą q ¨ ´ p ą q ¨ ´ p ą q ¨ ´ p ą q ¨ ´ p ą q ¨ . A similar replacement now needs to be done for the remaining half-shuffleproducts. As an example of how this works, consider only the terms with at the end p ą q ¨ ` p ą q ¨ ´ p ą q ¨ ´ p ą q ¨ “ ¨ ¨ ` p (cid:1) q ¨ ¨ ` ¨ ¨ ` p (cid:1) q ¨ ¨ ´ ¨ ¨ ´ p (cid:1) q ¨ ¨ ´ ¨ ¨ ´ p (cid:1) q ¨ ¨ “ ¨ ¨ ` ¨ ¨ ´ ¨ ¨ ´ ¨ ¨ , imilarly, applying this procedure for all the other terms, we obtain ‹ “ ` ¨ ¨ ` ¨ ¨ ` ¨ ¨ ` ¨ ¨ ` ¨ ¨ ` ¨ ¨ ` ¨ ¨ ` ¨ ¨ ` ¨ ¨ ` ¨ ¨ ` ¨ ¨ ` ¨ ¨ ` ¨ ¨ ` ¨ ¨ ` ¨ ¨ ` ¨ ¨ ` ¨ ¨ ` ¨ ¨ ´ ¨ ¨ ´ ¨ ¨ ´ ¨ ¨ ´ ¨ ¨ ´ ¨ ¨ ´ ¨ ¨ ´ ¨ ¨ ´ ¨ ¨ ´ ¨ ¨ ´ ¨ ¨ ´ ¨ ¨ ´ ¨ ¨ ´ ¨ ¨ ´ ¨ ¨ ´ ¨ ¨ ´ ¨ ¨ ´ ¨ ¨ ´ ¨ ¨ , as desired.Proof of Proposition 3.5. The second equality is obtained as follows, ÿ τ P S d sign p τ q ˆ ÿ σ P S d d ź i “ σ p i q τ p σ ` i q ˘˙ “ ÿ τ,σ P S d sign τ d ÿ i “ σ p i q τ p σ p i qq“ ÿ τ,σ P S d sign σ sign τ ˝ σ d ÿ i “ σ p i q τ ˝ σ p i q“ ÿ η,ρ P S d sign η sign ρ d ÿ i “ η p i q ρ p i q “ inv ¨˚˚˚˚˝ ... ... ´ ˛‹‹‹‹‚ . By Lemma 3.3, we have det (cid:1) p W d q “ ÿ σ P S d sgn p σ q (cid:1) di “ i σ p i q“ ÿ σ,τ P S d sign p σ q sign p τ q r σ p q τ p q ą σ p q τ p q ą . . . ą σ p d q τ p d qs . For brevity we write d ´ : “ d ´ and drop the brackets from the permu-tation action. We have, using (3), pp . . . q ą r τ d ´ στ d ´ sq ą r τ d στ d s“ ppp . . . q ą r τ d ´ στ d ´ sq (cid:1) τ d q ¨ στ d “ pp . . . q ą r τ d ´ στ d ´ sq ¨ τ d ¨ στ d ` pp . . . q (cid:1) τ d ´ (cid:1) τ d q ¨ στ d ´ ¨ στ d . σ and τ , we can choose two unique permutations τ and σ asfollows τ i “ $&% τ i , ď i ď d ´ τ d , i “ d ´ τ d ´ , i “ d and ˜ σ i “ $&% στ d ´ , i “ dστ d , i “ d ´ σ i , otherwise.Then, we have that ppp τ σ τ ą r τ σ τ sq ą .. q ą r τ d ´ σ τ d ´ sq ą r τ d σ τ d s“ ppp τ στ ą r τ στ sq ą .. q ą r τ d στ d ´ sq ą r τ d ´ στ d s“ pp . . . q ą r τ d στ d ´ sq ¨ τ d ´ ¨ στ d ` pp . . . q (cid:1) τ d (cid:1) τ d ´ q ¨ στ d ´ ¨ στ d . Thus, sign p σ q pp . . . q ą r τ d ´ στ d ´ sq ą r τ d στ d s` sign p σ q ppp τ σ τ ą r τ σ τ sq ą .. q ą r τ d ´ σ τ d ´ sq ą r τ d σ τ d s“ sign p σ q pp . . . q ą r τ d ´ στ d ´ sq ¨ τ d ¨ στ d ` sign p σ q pp . . . q ą r τ d ´ σ τ d ´ sq ¨ τ d ¨ σ τ d . Hence ÿ σ,τ P S d sgn p σ q (cid:1) di “ i σ p i q “ ÿ σ,τ P S d sgn p σ qpp . . . q ą r τ d ´ στ d ´ sq ¨ τ d ¨ στ d . Now, for any fixed value of τ d “ a and στ d “ b we can repeat the proceduredescribed above for ÿ σ,τ P S d τ d “ a ,στ d “ b sgn p σ qpp . . . q ą r τ d ´ στ d ´ sq ¨ τ d ¨ στ d “ ¨˚˚˝ ÿ σ,τ P S d τ d “ a ,στ d “ b sgn p σ qpp . . . q ą r τ d ´ στ d ´ sq ˛‹‹‚ ¨ a ¨ b “ ÿ σ,τ P S d τ d “ a ,στ d “ b sgn p σ qp . . . q ¨ τ d ´ ¨ στ d ´ s ¨ a ¨ b . In other words, step by step we can replace the half-shuffle product by con-catenation to obtain ÿ σ,τ P S d sgn p σ q (cid:1) di “ i σ p i q “ ÿ σ P S d sgn p σ q ˆ ÿ τ P S d d ź i “ τ p i q σ p τ ` i q ˘˙ . Main result
The aim of this section is to prove the following theorem.
Theorem 4.1.
For d ě , det (cid:1) p W d q “ inv ¨˚˚˚˚˝ ... ... d ´ d ˛‹‹‹‹‚ “ d inv ¨˚˚˚˚˝ ... d ˛‹‹‹‹‚ (cid:1) , i.e., using the notation (5) , det (cid:1) p W d q “ inv p t ,d q “ d inv p t ,d q (cid:1) . Before presenting the proof for Theorem 4.1, we show that Theorem 1.1 is aconsequence of it.
Corollary 4.2.
Equation (1) holds.Proof. det ¨˚˚˚˚˝ż dX p q dX p q . . . ż dX p q dX p d q ... . . . ... ż dX p d q dX p q . . . ż dX p q dX p d q ˛‹‹‹‹‚ “ det ¨˚˝ x , σ p X qy . . . x , σ p X qy ... . . . ... x d1 , σ p X qy . . . x dd , σ p X qy ˛‹‚ (4) “ A det (cid:1) p W d q , σ p X q E Thm . “ A d inv p t ,d q (cid:1) , σ p X q E (4) “ A d inv p t ,d q , σ p X q E “ d ˜ ÿ σ P S d sign p σ q ż dX σ p q . . . dX σ p d q ¸ . From Proposition 3.5 we already know that the first equality holds. To showthe second equality we use the following strategy:
Step 1
Show that the two terms lie in the same isotypic component of a repre-sentation of a certain subgroup of S d , and that this isotypic componenthas dimension 1. Step 2
Determine the pre-factors, by looking at the factors on a particularword in each side of the equality.14or
Step 1 , we first observe that both inv p t ,d q and inv p t ,d q (cid:1) are in the irre-ducible representation for S d corresponding to the shape p , , . . . , q .Denoteby χ the irreducible character for this shape.Define the subgroup of S d given by H : “ A p , q , p , q , . . . , p d ´ , d ´ q , p , q , p , q , . . . , p d ´ , d q E . (8)Note that χ is also a character for H , but it is not irreducible. Let χ p h q : “ sign p h q @ h P H, be the character for the one-dimensional sign-representation restricted from S d to H .Recall the two standard Young tableaux defined in (5). The following tworesults show that both p inv p t ,d q (cid:1) and inv p t ,d q lie in the isotypic componentrelated of this sign-representation. Lemma 4.3.
For all h P H , h ¨ inv ¨˚˚˚˚˝ ... d ˛‹‹‹‹‚ (cid:1) “ sign p h q ¨ inv ¨˚˚˚˚˝ ... d ˛‹‹‹‹‚ (cid:1) . Proof.
Without loss of generality we can suppose that h is the transposition p i, i ` q P H. The factor inv p t ,d q (cid:1) can be seen as a sum of words in T d p R d q with coeffi-cients in t˘ u . Then, each word would be the concatenation of letter comingfrom the left or right factor inv p t ,d q . For each word w , we label by L ˚ thepositions occupied by letters belonging to the left factor inv p t ,d q and by R ˚ the positions occupied by the letters belonging to the right factor inv p t ,d q .Denote by w p the letter situated on position p in the word w , and by p h w q p the letter situated on position p after applying the transposition h to w .We illustrate in Table 1 the possible cases for each word w .Since inv p t ,d q is totally antisymmetric, in each case presented in Table 1,the sign in front of the corresponding word after applying h changes due tothe transpositions of the indices of L ˚ , respectively R ˚ .It is immediate that inv p t ,d q satisfies the analogous statement.15able 1: Configurations of letters in a word w after applying the transposi-tion h “ p i, i ` q induces a change of sign. w i w i ` w i ` p h w q i p h w q i ` p h w q i ` L j L j ` L j ` L j ` L j ` L j L j L j ` R k R k L j ` L j L j R k R k ` R k ` R k L j R k R k ` R k ` R k ` R k ` R k R k R k ` L j L j R k ` R k R k L j L j ` L j ` L j R k R k L j R k ` R k ` L j R k L j R k L j ` L j ` R k L j Lemma 4.4.
For all h P H , h ¨ inv ¨˚˚˚˚˝ ... ... d ´ d ˛‹‹‹‹‚ “ sign p h q ¨ inv ¨˚˚˚˚˝ ... ... d ´ d ˛‹‹‹‹‚ . Now, we want to see that the dimension of this isotypic component is 1. Thisfollows from the fact that the multiplicity of the irreducible character χ in χ is equal to one in the case we are considering. Lemma 4.5.
Consider the decomposition of the representation (with re-spect to S d ) corresponding to χ into irreducible representations (with re-spect to H ). In this decomposition, the multiplicity of the irreducible sign-representation, i.e. the irreducible representation corresponding to χ , hasmultiplicity one. This result follows from the Littlewood-Richardson rule, which computesthe multiplicity of an irreducible representation on another representationby counting standard Young tableaux of some particular shape and content.This is a standard computation in representation theory of finite groups,see [FH91, Sta99] for more details.We are now ready to finish the proof of our main theorem.
Proof of Theorem 4.1.
Let us start by summarising the situation. By Propo-sition 3.5 we have that det (cid:1) p W d q “ inv p t ,d q . Now both inv p t ,d q and p inv p t ,d qq (cid:1) live in the irreducible S d -representation corresponding to theshape p , , . . . , q . Restricting this representation to the subgroup H of S d and decomposing into irreducible representations of H , we have that16 et (cid:1) p W d q and p inv p t ,d qq (cid:1) lie in the component corresponding to the sign-representation, see Lemmas 4.3 and 4.4. We also know, by Lemma 4.5, thatthis component is -dimensional.Our last step is to prove that the prefactors are as stated. Now, the word . . . dd can only be obtained from shuffling . . . d with . . . d , whichresults in a factor of d . Using the fact that it reduces to shuffle productsof blocks of letters (Proposition 3.5), one sees that in det (cid:1) p W d q , the word . . . dd appears with the factor one.The following example illustrates the computation of the prefactors fromTheorem 4.1 for d “ . Example 4.6.
The word appearsing in inv p t , q (cid:1) , can only comefrom the shuffle (cid:1) . This can happen in “ ways, namely (usingnotation from the proof of Lemma 4.3) L R L R L R , L R R L L R , L R L R R L , L R R L R L , R L L R L R , R L R L L R , R L L R R L , R L R L R L .On the other hand, in det (cid:1) p W q it only appears once in the term comingfrom (cid:1) (cid:1) (the diagonal of the matrix in the Leibniz formula for thedeterminant). Before stating and proving de Bruijn’s formula, we recall some definitionsand present some technical results. Let A be a commutative algebra over R . Lemma 5.1.
Let A P A d ˆ d skew-symmetric, v P A d and λ P A .If d is even det r A ` λvv J s “ det r A s . If d is odd det r A ` λvv J s “ d ÿ i “ det r R i s , where R i denotes the matrix A , with i th column by that row of λvv J Proof.
Let V : “ λvv J . For I Ă r n s : “ t , . . . , n u , let R I be the matrix A with the rows corresponding to I replaced by the corresponding rows of V .17orrespondingly, let C I be the matrix A with the columns corresponding to I replaced by the corresponding columns of V . Then det r A ` V s “ ÿ I Ăr n s det p R I q “ det p A q ` n ÿ i “ det p R t i u q det r A ` V s “ ÿ I Ăr n s det p C I q “ det p A q ` n ÿ i “ det p C t i u q , where we used the fact that det p R I q “ det p C I q “ if | I | ě . If d is odd, det r A s “ , since A is anti-symmetric. This yields the statement in this case,with ( R i : “ R t i u ).If d is even, we add the two expansions to get det r A ` V s “ det r A s ` n ÿ i “ ` det p R t i u q ` det p C t i u q ˘ . Since d is even, we get det p R t i u q “ ´ det p C t i u q , for all i P r n s . This finishesthe proof.For the following statement recall the matrix W d from (6) with entries inthe shuffle algebra T p R d q . Lemma 5.2.
Write
Sym r W d s : “ p W d ` W J d q . Then we have
Sym r W d s “ ¨˚˚˚˝ ... d ˛‹‹‹‚ (cid:1) ¨˚˚˚˝ ... d ˛‹‹‹‚ J . Proof.
We have ¨˚˚˚˝ ... d ˛‹‹‹‚ (cid:1) ¨˚˚˚˝ ... d ˛‹‹‹‚ J “ ¨˚˝ (cid:1) (cid:1) . . . (cid:1) d ... ... . . . ... d (cid:1) (cid:1) . . . d (cid:1) d ˛‹‚ “ ¨˚˝ `
11 12 ` . . . ` d1 ... ... . . . ... d1 `
1d d2 ` . . . dd ` dd ˛‹‚ “ W d ` W J d . Definition 5.3.
Let A “ p a ij q P A d ˆ d be a skew-symmetric matrix, where d is even. Then Pf A r A s : “ d { p d { q ! ÿ π P S d sign p π q d { ź i “ a π p i ´ q a π p i q is called the Pfaffian of A .For the following statement see for instance [Art57, pages 140-141], [Led93],and references therein. Lemma 5.4.
Let d be even and A P A d ˆ d a skew-symmetric matrix. Wehave the following identity: det A r A s “ p Pf A r A sq . The de Bruijn’s formula has two different formulations depending on theparity of d . We compare both formulations with our results.We recall that any matrix M can be written as the sum of its symmetricpart and its anti-symmetric part M “ Sym r M s ` Anti r M s , with Sym r M s : “ p M ` M J q , Anti r M s : “ p M ´ M J q . For d even, de Bruijn’s formula ([dB55], see also [DR19, Remark 3.18]),reads, in modern language, as follows. Theorem (de Bruijn’s formula - Even case) . inv ¨˚˚˚˚˝ ... d ˛‹‹‹‹‚ “ d { Pf (cid:1) r Anti r W d ss . Proof.
We deduce this from our results in the main text. First, Theorem 4.1gives inv ¨˚˚˚˚˝ ... d ˛‹‹‹‹‚ (cid:1) “ d det (cid:1) r W d s . det (cid:1) r W d s “ det (cid:1) r Anti r W d ss . By properties of the Pfaffian, Lemma 5.4, det (cid:1) r Anti r W d ss “ Pf (cid:1) r Anti r W d ss (cid:1) . Hence inv ¨˚˚˚˚˝ ... d ˛‹‹‹‹‚ (cid:1) “ d Pf (cid:1) r Anti r W d ss (cid:1) . Since the shuffle algebra is commutative and integral, we deduce that: inv ¨˚˚˚˚˝ ... d ˛‹‹‹‹‚ “ ˘ d { Pf (cid:1) r Anti r W d ss . By comparing the coefficient of the word . . . d , we deduce that the signmust be positive, and have thus shown de Bruijn’s formula in the even case. Example 5.5. de Bruijn’s formula in the case d “ gives inv ¨˚˚˚˚˝ ˛‹‹‹‹‚ “ (cid:1) ¨˚˚˝ ´
21 13 ´
31 14 ´ ´ ´
32 24 ´ ´
13 32 ´ ´ ´
14 42 ´
24 43 ´ ˛‹‹‚ , whereas Theorem 4.1 gives inv ¨˚˚˚˚˝ ˛‹‹‹‹‚ (cid:1) “
16 det (cid:1) ¨˚˚˝
11 12 13 1421 22 23 2431 32 33 3441 42 43 44 ˛‹‹‚ . Now, we look at the odd case. 20 heorem (de Bruijn’s formula - Odd case) . For d odd, de Bruijn’s formula([dB55]) reads as inv ¨˚˚˚˚˝ ... d ˛‹‹‹‹‚ “ Pf (cid:1) r Z d s , where Z d denotes the matrix of the form Z d “ ¨˚˚˚˚˚˝ r W d s ... d ´ ´ ¨ ¨ ¨ ´ d ˛‹‹‹‹‹‚ . Proof.
We follow a similar strategy as in the even-dimensional case, to showthis theorem using our results. As before, we decompose into symmetric andanti-symmetric part, W d “ Sym r W d s ` Anti r W d s . By Lemma 5.2,
Sym r W d s “ vv J for v J “ “ ¨ ¨ ¨ d ‰ . Then byLemma 5.1 det (cid:1) r W d s “ d ÿ i “ det (cid:1) r R i s , where R i is the matrix Anti r W d s with i th row replaced by v i v J . Now d ÿ i “ det (cid:1) r R i s “ ´ d det (cid:1) ¨˚˚˚˚˚˝ r W d s ... d ´ ´ ¨ ¨ ¨ ´ d ˛‹‹‹‹‹‚ “ ´ d det (cid:1) p Z d q , which gives d det (cid:1) r W d s “ det (cid:1) p Z d q . Combining with our Theorem 4.1, weget inv ¨˚˚˚˚˝ ... d ˛‹‹‹‹‚ (cid:1) “ d det (cid:1) r W d s “ det (cid:1) p Z d q .
21y Lemma 5.4 (note that Z d is even-dimensional), det (cid:1) r Z d s “ Pf (cid:1) r Z d s (cid:1) . Hence inv ¨˚˚˚˚˝ ... d ˛‹‹‹‹‚ (cid:1) “ Pf (cid:1) r Z d s (cid:1) . As for the even case, since the shuffle algebra is commutative and integral,and by comparing the coefficient of the word . . . d , we deduce de Bruijn’sformula in the odd case. Example 5.6. de Bruijn’s formula in the case d “ gives inv ¨˚˚˝ ˛‹‹‚ “ Pf (cid:1) ¨˚˚˝ ´
21 13 ´
31 121 ´ ´
32 231 ´
13 32 ´ ´ ´ ´ ˛‹‹‚ whereas Theorem 4.1 gives inv ¨˚˚˝ ˛‹‹‚ (cid:1) “ (cid:1) ¨˝
11 12 1321 22 2331 32 33 ˛‚ , which, by the preceding argument, is equal to det (cid:1) ¨˚˚˝ ´
21 13 ´
31 121 ´ ´
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