q-alg/ v2 15 Sep 1997
|
|
- Meagan Bradley
- 5 years ago
- Views:
Transcription
1 DOMINO TABLEAUX, SCH UTZENBERGER INVOLUTION, AND THE SYMMETRIC GROUP ACTION ARKADY BERENSTEIN Department of Mathematics, Cornell University Ithaca, NY 14853, U.S.A. q-alg/ v 15 Sep 1997 ANATOL N. KIRILLOV CRM, University of Montreal C.P. 618, Succursale A, Montreal (Quebec) H3C 3J7, Canada and Steklov Mathematical Institute, Fontanka 7, St.Petersburg , Russia ABSTRACT We dene an action of the symmetric group S [ n ] on the set of domino tableaux, and prove that the number of domino tableaux of weight 0 does not depend on the permutation of the weight 0. A bijective proof of the well-known result due to J. Stembridge that the number of self{evacuating tableaux of a given shape and weight = ( 1 ; : : : ; n+1 [ ]; [ n ] ; : : : ; 1 ), is equal to that of domino tableaux of the same shape and weight 0 = ( 1 ; : : : ; [ n+1 ]) is given. 0. Introduction Domino tableaux play a signicant role in the representation theory of the symmetric group S n, and the theory of symmetric polynomials. The shortest denition of semistandard domino tableaux is given in [10], p In particular, the generating function of the number of domino tableaux of shape = ( 1 n 0), and weight 0 = ( 1 ; : : : ; n+1 [ ]) is given by specializing the Schur polynomial s (x) at x 7! y where y n =?y n?1, y n? =?y n?3 ; : : :: s (y) = X 0 K () ; 0 y 1 n y n? ; (0:1) where the sign depends only on. Recently they were studied in connection with the representation theory of GL n. In particular, in [1] John Stembridge proved the following conjecture by Richard Stanley (who stated it for 0 = (1; 1; : : : ; 1), that is, for standard tableaux). Theorem 0.1 ([1]). The number of self-evacuating tableaux of shape and weight = ( 1 ; : : : ; n+1 [ ]; [ n ] ; : : : ; 1 ), equals the number of domino tableaux K () ; of shape 0 and weight 0.
2 A. Berenstein and A.N. Kirillov The self-evacuating tableaux are dened as xed points of the Schutzenberger evacuation involution S on the semi-standard tableaux (see [11], [1]). The proof of Theorem 0.1 in [1] uses properties of the canonical basis for the GL n -module V. A key ingredient of the proof is the result of [3], Theorem 8., which states that S is naturally identied with the longest element of the symmetric group S n acting on the canonical basis of V. There are combinatorial algorithms (see e.g., [5] or [9]) which lead to a bijective proof of Theorem 0.1 for the standard tableaux. For semi-standard tableaux and even n a similar algorithm appeared in [7], remark on the page 399. In the present paper we give a bijective proof of Theorem 0.1 for any n (Theorem 1.). The main idea of the bijection constructed in this paper is that (semi-standard) domino tableaux are naturally identied with those ordinary (semi-standard) tableaux which are xed under a certain involution D (see the Appendix). We discover that the involutions D and S are conjugate as automorphisms of all tableaux, say S = PD (see Lemma 1.3). Thus, the conjugating automorphism P bijects the domino tableaux onto the self-evacuating tableaux. The main dierence of our construction and those given in [5], [9] and [7] is that instead of a description of combinatorial algorithms as it was done in the papers mentioned, we give a direct algebraic formula for the bijection under consideration. Our approach is based on results from []. It follows from (0.1) that the number K () ; does not depend on the permutation of 0 the weight 0. This is analogous to the property of the ordinary Kostka numbers K ;. Recently, Carre and Leclerc constructed in [4] an appropriate action of the symmetric group S n (for even n) which realizes this property. Their construction used a generalized RSK-type algorithm ([4], Algorithm 7.1, and Theorem 7.8). The second result of the present paper is an algebraic construction of action of the symmetric group S [ n ] on Tab D n. This action is parallel to the action of S n on the ordinary tableaux studied in [], and has a natural interpretation in terms of self{evacuating tableaux (Theorem 1.8). One can conjecture that for even n our action coincides with the Carre{Leclerc's action. The material of the paper is arranged as follows. In Section 1 we list main results and construct the bijection. Section contains proofs of Theorems 1.6 and 1.8 on the action of the symmetric group S [ n ]. For the reader's convenience, in Appendix we remind denitions of the Bender{Knuth's involution and domino tableaux. Acknowledgements. The authors are grateful to George Lusztig for fruitful discussions. Special thanks are due to Mark Shimozono for an inestimable help in the preparation of the manuscript. Both authors are grateful to CRM and the Department of Mathematics of University of Montreal, for hospitality and support during this work.
3 1. Main results Domino tableaux, Schutzenberger involution, and the symmetric group action 3 Following [10], denote by T ab n the set of all (semi-standard Young) tableaux with entries not exceeding n. For a partition with at most n rows, and for an integer vector = ( 1 ; : : : ; n ) denote by T ab () the set of T T ab n of shape and weight (for the reader's convenience we collect all necessary denitions in the Appendix). Let t i : T ab n! T ab n, (i = 1; : : : ; n? 1) be the Bender-Knuth automorphisms which we dene in the Appendix (see also [1],[] and [6]). Dene the automorphism D : T ab n! T ab n by the formula: D = D n := t n?1 t n?3 ; (1:1) i.e., D = t n?1 t n?3 t for odd n, and D = t n?1 t n?3 t 1 for even n. Denote by of T ab D n the set of tableaux T T ab n such that D(T ) = T. For every shape and weight 0 = ( 1 ; : : : ; n+1 [ ]) denote T abd (0 ) = T ab () \ T ab D n, where = ( n+1 [ ]; ; k; k ; ; ; ; 1 ; 1 ). The semi-standard domino tableaux of shape and weight 0 were studied by several autors (see, e.g., [10], p. 139). We also dene them in the Appendix. Our rst "result" is another denition of domino tableaux. Proposition-Denition 1.1. For every and 0 as above the domino tableaux of shape and weight 0 are in a natural one-to-one correspondence with the set T ab D (0 ). We prove this proposition in the Appendix. From now on we identify the domino tableaux with the elements of T ab D n. Let us introduce the last bit of notation prior to the next result. Let p i := t 1 t t i for i = 1; : : : ; n? 1 (this p i is the inverse of the i-th promotion operator dened in [11]). Dene the automorphism P : T ab n! T ab n by the formula: P = P n := p n?1 p n?3 ; (1:) so P = p n?1 p n?3 p for odd n, and P = p n?1 p n?3 p 1 for even n. It is well-known (see, e.g., [],[6]) that the Schutzenberger evacuation involution factorizes as follows. S = S n = p n?1 p n? p 1 (1:3) We regard (1.3) as a denition of S. Denote by T ab S n T ab n the set of the self-evacuating tableaux, that is all the tableaux T T ab n xed under S. Similarly, denote T ab S () = T ab () \ T ab S n. It is well-known that this set is empty unless i = n+1?i for i = 1; ; : : : ; [ n+1 ].
4 4 A. Berenstein and A.N. Kirillov Theorem 1.. The above automorphism P of T ab D n induces the bijection T ab D n = T ab S n : (1:4) More precisely, P induces the bijection T ab D (0 ) = T ab S () for any shape and the weight 0 = ( 1 ; : : : ; n+1 [ ]), where = ( 1; : : : ; n+1 [ ]; [ n ] ; : : : ; 1 ). Remark. Theorem 1. is the "realization" of the theorem in Section 0. The proof of this theorem is so elementary that we present it here. Proof of Theorem 1.. The statement (1.4) immediately follows from the surprisingly elementary lemma below. Lemma 1.3. S = PD. Indeed, if T T ab D n then T = D(T ) = SP(T ) by Lemma 1.3, so S(P(T )) = P(T ), that is, P(T ) T ab S n. This proves the inclusion P(T abd n ) T abs n. The opposite inclusion also follows. Remark. By denition (see the Appendix), the involutions t 1 ; : : : ; t n?1 involved in Lemma 1.3, satisfy the following obvious relations (see e.g. []): t i = id; t it j = t j t i ; 1 i; j n? 1; ji? jj > 1 : (1:5) In fact, the factorization in Lemma 1.3 is based only on this relations (as we can see from the proof below), and, hence, makes sence in any group with the relations (1.5). It also follows from the denition of t i that t i : T ab () = T ab ((i; i + 1)() : This proves the second assertion of Theorem 1. provided that Lemma 1.3 is proved. Proof of Lemma 1.3. We will proceed by induction on n. If n = then S = t 1 while P = t 1, and D = t 1. If n = 3 then S 3 = t 1 t t 1 while D = t, and P = t 1. Then, for any n > 3 the formula (1.3) can be rewritten as S n = p n?1 S n?1. It is easy to derive from (1.5) that S n = S n?1 p?1 n?1. Applying both of the above relations, we obtain S n = p n? S n? p?1 n?1 : (1:6) Finally, by (1.5) and the inductive assumption for n?, S n = p n? S n? p?1 = n?1 p n?p n? D n? n? p?1 = n?1 p n?p n? t n?1 D n n? p?1 n?1 since D n? = t n?1 D n. Finally, n? p?1 = n?1 P?1 n, and p n? P n? t n?1 = p n? t n?1 P n? = p n?1 P n? = P n. Lemma 1.3 is proved. Theorem 1. is proved. Recall (see the Appendix) that the weight of a domino tableau T T ab D n is the vector 0 = ( 1 ; : : : ; n+1 [ ]), where 1 is the number of occurences of n, is that of n?, and so on. Denote by T ab D n ( 0 ) the set of all T T ab D n of weight 0. Our next main result is the following
5 Domino tableaux, Schutzenberger involution, and the symmetric group action 5 Theorem 1.4. There exists a natural faithful action of the symmetric group S [ n ] on T ab D n preserving the shape and acting on weight by permutation. In order to dene the action precisely, let us recall one of the results of []. Theorem 1.5 ([]). There is an action of S n on T ab n given by (i; i + 1) 7! s i, where s i = S i t 1 S i ; i = 1; : : : ; n ; (1:7) that is, the s i are involutions, and (s j s j+1 ) 3 = id; s i s j = s j s i for all i; j such that jj?ij > 1. (Here S i given by (1.3) acts on T ab n ). Denition. For n 4 dene the automorphisms i : T ab n! T ab n, i = ; : : : ; n? by the formula i := t i s i?1 s i+1 t i : (1:8) Remark. The automorphisms s i were rst dened by Lascoux and Schutzenberger in [8] in the context of plactique monoid theory. The denition (1.7) of s i rst appeared in [] in a more general situation. As a matter of fact, for the restriction of s i to the set of tableaux, (1.7) is implied in [8]. Theorem 1.4 follows from a much stronger result, namely the following theorem: Theorem 1.6. (a) D i = i D if and only if i n (mod ). (b) The automorphisms ; 3 ; n? are involutions satisfying the Coxeter relations ( i j ) n ij = id for i; j = ; : : : ; n? ; where n ij = 8< : 3 if jj? ij = 1 or 6 if jj? ij = 3 if jj? ij > 3 : (1:9) We will prove Theorem 1.6 in Section. Remark. One can conjecture that the relations (1.8) (together with i presentation of the group n generated by the i. Indeed, the action of S [ n ] on T ab n from Theorem 1.4 can be dened by (i; i + 1) 7! n?i ; i = ; : : : ; n? : = id) give a According to Theorem 1.6(a), this action preserves T ab D n. By denition (1.7), this action preserves shape of tableaux and permutes the weights as follows: n?i : T ab D ( 0 ) = T ab D ((i; i + 1)( 0 )) : (1:10) Let us try to get the action of S [ n ] on T ab S n using the following nice property of the involutions s 1 ; : : : ; s n?1 (which can be found, e.g., in [], Proposition 1.4): Ss i S = s n?i ; i = 1; : : : ; n? 1 : (1:11) Let 1 ; : : : ; [ n ]?1 be dened by k := s k s n?k. The following proposition is obvious.
6 6 A. Berenstein and A.N. Kirillov Proposition 1.7. (a) S k = k S for all k. (b) The group generated by the k is isomorphic to S [ n ] via (k; k + 1) 7! k ; k = 1; : : : ; [ n ]? 1 : (1:1) Restricting the action (1.1) to T ab S n we obtain the desirable action of S [ n ] on T ab S n. Our last main result compares of the latter action with that on T ab D n. The following theorem conrms naturality of our choice of j in (1.8). Theorem 1.8. For k = 1; : : : ; [ n ]? 1, we have P?1 k P = n?k.. Proof of Theorems 1.6, 1.8 and related results We start with a collection of formulas relating s i and t j. First of all, s i t j = t j s i (:1) for all i; j with jj? ij > 1 (This fact follows from [8] and is implied in [], Theorem 1.1). Then, by denition (1.8) of i, i t j = t j i (:) whenever jj? ij >. Furthermore, (1.3) and (1.7) imply that More generally, Finally, (.3) and (.1) imply that s j = p j s j?1 p?1 j ; j = 1; : : : ; n? 1 : (:3) s i = p j s i?1 p?1 j ; 1 i j < n : (:4) t j?1 s j t j?1 = t j s j?1 t j : (:5) Proof of Theorem 1.6(a). Let i be congruent n modulo. Then by (.) and the denition (1.1) of D, D i D = t i+1 t i?1 i t i+1 t i?1 : (:6) We now prove that (.6) is equal to i. Indeed, D i D = t i+1 t i?1 i t i+1 t i?1 = t i+1 t i?1 t i s i?1 s i+1 t i t i+1 t i?1 :
7 Domino tableaux, Schutzenberger involution, and the symmetric group action 7 Now we will apply the relations (.5) in the form t i?1 t i s i?1 = s i t i?1 t i, and s i+1 t i t i+1 = t i t i+1 s i. We obtain D i D = t i+1 (s i t i?1 t i )(t i t i+1 s i )t i?1 = t i+1 s i t i?1 t i+1 s i t i?1 = t i+1 s i t i+1 t i?1 s i t i?1 : Applying (.5) with j = i? 1; i once again, we nally obtain D i D = (t i s i+1 t i )(t i s i?1 t i ) = t i s i+1 s i?1 t i = t i s i?1 s i+1 t i = i by Theorem 1.5, which proves that D i = i D. The implication "if" of Theorem 1.6(a) is proved. Let us prove the "only if" implication. Let be the canonical homomorphism from the group generated by t 1 ; : : : ; t n?1 to S n dened by t j 7! (j; j + 1). Then ( i ) = (i; i+1)(i?1; i)(i+1; i+)(i; i+1) for i = ; : : : ; n?, and (D) = (n?1; n)(n?3; n?). If i 6 n (mod ) then ( i ) and (D) do not commute. Hence i D 6= D i for such i. Thus Theorem 1.6(a) is proved. Proof of Theorem 1.6(b). We are going to consider several cases when j? i = 1; ; 3 and > 3. In each case we compute the product i j up to conjugation, and express it as a product of several s k. These products are always of nite order, because, according to Theorem 1.5, these s k 's generate the group isomorphic to S n. The easiest case is j?i > 3. Then all the ingredients of j in (1.7) (namely t j ; s j?1 ; s j+1 ) commute with those of i (namely t i ; s i?1 ; s i+1 ). Thus j commutes with i, that is, ( i j ) = id, and we are done in this case. In what follows we will write a b if a is conjugate to b. Case j? i = 3, i.e., j = i + 3. We have i i+3 = t i s i?1 s i+1 t i t i+3 s i+ s i+4 t i+3 s i?1 s i+1 s i+ s i+4 by the above commutation between remote s k and t l. By Theorem 1.5, ( i i+3 ) 6 (s i?1 ) 6 (s i+1 s i+ ) 6 (s i+4 ) 6 = id. So ( i i+3 ) 6 = id and we are done with this case. Case j? i =, i.e., j = i +. We have i i+ = t i s i?1 s i+1 t i t i+ s i+1 s i+3 t i+ t i+ t i s i?1 s i+1 t i t i+ s i+1 s i+3 = t i s i?1 t i+ s i+1 t i+ t i s i+1 s i+3 = t i s i?1 t i+1 s i+ t i+1 t i s i+1 s i+3 by (.5) with j = i +. Conjugating the latter expression with t i, we obtain, i i+ s i?1 t i+1 s i+ t i+1 t i s i+1 s i+3 t i = s i?1 t i+1 s i+ t i+1 t i s i+1 t i s i+3 = s i?1 t i+1 s i+ t i+1 t i+1 s i t i+1 s i+3 = s i?1 t i+1 s i+ s i t i+1 s i+3
8 8 A. Berenstein and A.N. Kirillov t i+1 s i?1 t i+1 s i+ s i t i+1 s i+3 t i+1 = s i?1 s i+ s i s i+3 = s i?1 s i s i+ s i+3 : Thus, ( i i+ ) 3 (s i?1 s i s i+ s i+3 ) 3 = (s i?1 s i ) 3 (s i+ s i+3 ) 3 = id, and we are done with this case too. Case j? i = 1, i.e., j = i + 1. We have i i+1 = t i s i?1 s i+1 t i t i+1 s i s i+ t i+1 = t i s i?1 s i+1 t i t i+1 s i t i+1 t i+1 s i+ t i+1 = t i s i?1 s i+1 t i t i s i+1 t i t i+ s i+1 t i+ = t i s i?1 t i t i+ s i+1 t i+ = t i?1 s i t i?1 t i+ s i+1 t i+ s i t i+ s i+1 t i+ s i s i+1 : Thus, ( i i+1 ) 3 (s i s i+1 ) 3 = id, and we are done with this nal case. This nishes the proof of Theorem 1.6. Proof of Theorem 1.8. First of all we prove Theorem 1.8 with k = 1, that is, Since P n = p n P n?, the left hand side of (.7) equals n s 1s n?1 P n = n? (:7) n? p?1 n?1 s 1p n?1 p?1 n?1 s n?1p n?1 P n? = n? (p?1 n?1 s 1p n?1 )s n? P n? : (:8) Now we prove by induction on n that p?1 n?1 s 1p n?1 = p n? s n?1 p?1 n? (:9) Indeed, if n =, (.9) is obvious (we agree that p 0 = t 0 = id). Then using (.9) as inductive assumption, we obtain p?1 n s 1p n = t n (p?1 n?1 s 1p n?1 )t n = t n (p n? s n?1 p?1 n? )t n = p n? t n s n?1 t n p?1 n? = p n?t n?1 s n t n?1 p?1 n? = p n?1s n p?1 n?1 by (.5), and we are done with (.9). Finally, in order to nish the proof of (.7), let us substitute (.9) into (.8). Thus, taking into account that P n? = p n?3 P n?4 = p n? t n? P n?, (.8) is equal to n? (p n?s n?1 p?1 n? )s n?p n? = n?4 t n?p?1 n? (p n?s n?1 p?1 n? )s n?p n? t n? P n?4 = n?4 t n?s n?1 (p?1 n? s n?p n? )t n? P n?4 = n?4 t n?s n?1 s n?3 t n? P n?4 which proves Theorem 1.8 for k = 1. = t n? s n?1 s n?3 t n? = n? ;
9 Domino tableaux, Schutzenberger involution, and the symmetric group action 9 Now using induction on n we will prove the general case, namely, the formula n s ks n?k P n = n?k ; 1 k [ n ]? 1: (:10) Indeed, n s ks n?k P n = n? (p?1 n?1 s ks n?k p n?1 )P n? : (:11) But p?1 n?1 s ks n?k p n?1 = (p?1 n?1 s kp n?1 )(p?1 n?1 s n?kp n?1 ) = s k?1 s n?k?1 by (.4) applied with j := n? 1; i = k, and i = n? k. Thus (.11) is equal to n? s k?1s n?k?1 P n? = n??(k?1) = n?k by the inductive assumption (.10) with n?. Theorem 1.8 is proved. 3. Appendix. Denitions of domino tableaux By denition from [10], a (semi-standard Young) tableau T with entries not exceeding n is an ascending sequence of Young diagrams T = (f;g = 0 1 : : : n ) such that i = i n i?1 is a horizontal strip for i = 1; : : : ; n (we allow i+1 = i ). Denote by T ab n the set of all (semi-standard Young) tableaux with entries not exceeding n. Dene the shape of T by = n, and the weight = ( 1 ; ; : : : ; n ) of T by i = j i j for i = 1; : : : ; n. Denote by T ab n () the set of all T T ab n having weight. One visualizes the tableau T as lling of each box of i with the number i for i = 1; : : : ; n. Let T = ( 0 1 ) be a skew -tableau, that is, 1 n 0 and n 1 both are horizontal strips. For such a skew -tableau T dene the skew -tableau t(t ) = T = ( ) as follows. First of all we visualize T as the shape D = n 0 lled with the letters a and b where the letters a ll the horizontal strip 1 n 0, and the letters b ll the remaining horizontal strip n 1. Then t(t ) will be a new lling of D with the same letters a and b which we construct below. Let D k be the longest connected sub-row of the k-th row of D, whose boxes have no horizontal edges in common with other boxes of D. D k contains say, l boxes on the left lled with a, and r boxes (on the right) lled with b. We will replace this lling of D k with the r letters a on the left, and l letters b on the right. We do this procedure for every k, and leave unchanged the lling of the complement of the union of all the D k in D. It is easy to see that the new lling of D is a skew -tableau of the form t(t ) = T = ( ). Clearly the correspondence T 7! t(t ) is an involutive automorphism of the set of skew -tableaux. The following lemma is obvious.
10 10 A. Berenstein and A.N. Kirillov Lemma-Denition A1. Let D be a skew Young diagram. Then the following are equivalent: (i) There is a (unique) skew -tableau T = ( 0 1 ) with D = n 0 such that t(t ) = T (ii) There is a (unique) covering of D by (non-overlapping) dominoes such that every domino has no common horizontal edges with other boxes of D. Dene the Bender-Knuth involution ([1], see also []) t i : T ab n! T ab n as follows. T = ( i?1 i i+1 ) T ab n. Dene T i (T ) T ab n by t i (T ) = ( i?1 0 i i+1 ) where 0 i is dened by ( i?1 0 i i+1) = t( i?1 i i+1 ), and other j remain unchanged. Denote the resulting tableau by t i (T ). It is easy to see that the correspondence T 7! t i (T ) is a well-dened involutive automorphism t i : T ab n! T ab n, and t i t j = t j t i whenever jj? ij > 1. A domino tableau T with entries not exceeding n+1 is an ascending sequence of the Young diagrams T = ( " "+ n? n ) with " = n? [ n ], such that (i) " = ; if n is even, (i.e., " = 0), and " is a connected horizontal strip if n is odd, (i.e., " = 1); (ii) Each dierence D k = n?k n n?k+ is subject to condition (ii) of Lemma A1. The shape of T is dened to be n. We also dene the weight 0 = ( 1 ; : : : ; n+1 [ ]) of T by k = jd kj for k = 1; : : : ; [ n+1]. Thus the above denitions together with Lemma-Denition A1 make Proposition- Denition 1.1 obvious.
11 References Domino tableaux, Schutzenberger involution, and the symmetric group action 11 [1] E. Bender, D. Knuth, Enumeration of plane partitions. J. Combinatorial Theory Ser. A, 13 (197), 40{54. [] A. Berenstein, A.N. Kirillov, Groups generated by involutions, Gelfand-Tsetlin patterns and combinatorics of Young tableaux, Algebra i Analiz, 7 (1995), no. 1, 9{15; translation in St. Petersburg Math. J. 7 (1996), no. 1, 77{17. [3] A. Berenstein, A. Zelevinsky, Canonical bases for the quantum group of type A r and piecewise-linear combinatorics, Duke Math. J., 8 (1996), no. 3, 473{50. [4] C. Carre, B. Leclerc, Splitting the square of Schur function into symmetric and antisymmetric parts, J. Algebraic Comb., 4 (1995), n.3, 01{31. [5] S. Fomin, Generalized Robinson-Schensted-Knuth correspondence, Journal of Soviet Math., 41 (1988), 979{991. [6] E. Gansner, On the equality of two plane partition correspondences, Discrete Math., 30 (1980), 11{13. [7] A.N. Kirillov, A. Lascoux, B. Leclerc, J.-Y. Thibon, Series generatrices pour les tableaux de domino, C.R.Acad. Sci. Paris, 318 (1994), 395{400. [8] A. Lascoux, M.-P. Schutzenberger, Le monode plaxique. Noncommutative structures in algebra and geometric combinatorics (Naples, 1978), pp. 19{156, Quad. "Ricerca Sci.", 109, CNR, Rome, [9] M.A. van Leeuwen, The Robinson-Schensted and Schutzenberger algorithms, an elementary approach, Electronic Journal of Combinatorics, 3, n. (1996). [10] I. Macdonald, Symmetric functions and Hall polynomials. Second edition. With contributions by A. Zelevinsky. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, [11] M.-P. Schutzenberger, Promotion des morphismes d'ensembles ordonnes, Discrete Math., (197), 73{94. [1] J. Stembridge, Canonical bases and self-evacuating tableaux, Duke Math. J., 8 (1996), no.3, 585{606.
A RELATION BETWEEN SCHUR P AND S. S. Leidwanger. Universite de Caen, CAEN. cedex FRANCE. March 24, 1997
A RELATION BETWEEN SCHUR P AND S FUNCTIONS S. Leidwanger Departement de Mathematiques, Universite de Caen, 0 CAEN cedex FRANCE March, 997 Abstract We dene a dierential operator of innite order which sends
More informationarxiv:math/ v1 [math.co] 15 Sep 1999
ON A CONJECTURED FORMULA FOR QUIVER VARIETIES ariv:math/9909089v1 [math.co] 15 Sep 1999 ANDERS SKOVSTED BUCH 1. Introduction The goal of this paper is to prove some combinatorial results about a formula
More information2 Anatol N. Kirillov and Arkadiy D. Berenstein points set, (K n ) Z, of the cone K n is in a one-to-one correspondence with the set STY (n) of standar
GROUPS GENERATED BY INVOLUTIONS, GELFAND-TSETLIN PATTERNS AND COMBINATORICS OF YOUNG TABLEAUX ANATOL N. KIRILLOV RIMS, Kyoto University, Kyoto 66, Japan Steklov Mathematical Institute, Fontanka 27, St.Petersburg,
More information2 IGOR PAK so we loose some information about the structure of the tilings since there could be many tilings of with the same multiset of tiles (see e
RIBBON TILE INVARIANTS Igor Pak MIT E-mail: pak@math.mit.edu September 30, 1997 Abstract. Let T be a nite set of tiles, B be a set of regions tileable by T. We introduce a tile counting group G (T; B)
More informationarxiv: v1 [math.co] 25 May 2011
arxiv:1105.4986v1 [math.co] 25 May 2011 Gog and Magog triangles, and the Schützenberger involution Hayat Cheballah and Philippe Biane Abstract. We describe an approach to finding a bijection between Alternating
More informationCombinatorial Structures
Combinatorial Structures Contents 1 Permutations 1 Partitions.1 Ferrers diagrams....................................... Skew diagrams........................................ Dominance order......................................
More informationCylindric Young Tableaux and their Properties
Cylindric Young Tableaux and their Properties Eric Neyman (Montgomery Blair High School) Mentor: Darij Grinberg (MIT) Fourth Annual MIT PRIMES Conference May 17, 2014 1 / 17 Introduction Young tableaux
More informationTWO RESULTS ON DOMINO AND RIBBON TABLEAUX
TWO RESULTS ON DOMINO AND RIBBON TABLEAUX THOMAS LAM arxiv:math/0407184v1 [math.co] 11 Jul 2004 Abstract. Inspired by the spin-inversion statistic of Schilling, Shimozono and White [8] and Haglund et al.
More informationSCHUR FUNCTIONS, PAIRING OF PARENTHESES, JEU DE TAQUIN AND INVARIANT FACTORS
SCHUR FUNCTIONS, PAIRING OF PARENTHESES, JEU DE TAQUIN AND INVARIANT FACTORS OLGA AZENHAS For Eduardo Marques de Sá on his 0th birthday Abstract We translate the coplactic operation by Lascoux and Schützenberger
More informationq-alg/ Mar 96
Integrality of Two Variable Kostka Functions Friedrich Knop* Department of Mathematics, Rutgers University, New Brunswick NJ 08903, USA knop@math.rutgers.edu 1. Introduction q-alg/9603027 29 Mar 96 Macdonald
More informationCoxeter-Knuth Classes and a Signed Little Bijection
Coxeter-Knuth Classes and a Signed Little Bijection Sara Billey University of Washington Based on joint work with: Zachary Hamaker, Austin Roberts, and Benjamin Young. UC Berkeley, February, 04 Outline
More informationCoxeter-Knuth Graphs and a signed Little Bijection
Coxeter-Knuth Graphs and a signed Little Bijection Sara Billey University of Washington http://www.math.washington.edu/ billey AMS-MAA Joint Meetings January 17, 2014 0-0 Outline Based on joint work with
More informationMultiplicity-Free Products of Schur Functions
Annals of Combinatorics 5 (2001) 113-121 0218-0006/01/020113-9$1.50+0.20/0 c Birkhäuser Verlag, Basel, 2001 Annals of Combinatorics Multiplicity-Free Products of Schur Functions John R. Stembridge Department
More informationSplitting the Square of a Schur Function into its Symmetric and Antisymmetric Parts
Journal of Algebraic Combinatorics 4 (1995), 201-231 1995 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands. Splitting the Square of a Schur Function into its Symmetric and Antisymmetric
More informationA Generating Algorithm for Ribbon Tableaux and Spin Polynomials
Discrete Mathematics and Theoretical Computer Science DMTCS vol. 9:, 007, 5 58 A Generating Algorithm for Ribbon Tableaux and Spin Polynomials Francois Descouens Institut Gaspard Monge, Université de Marne-la-Vallée
More informationON SOME FACTORIZATION FORMULAS OF K-k-SCHUR FUNCTIONS
ON SOME FACTORIZATION FORMULAS OF K-k-SCHUR FUNCTIONS MOTOKI TAKIGIKU Abstract. We give some new formulas about factorizations of K-k-Schur functions, analogous to the k-rectangle factorization formula
More informationSergey Fomin* and. Minneapolis, MN We consider the partial order on partitions of integers dened by removal of
Rim Hook Lattices Sergey Fomin* Department of Mathematics, Massachusetts Institute of Technology Cambridge, MA 02139 Theory of Algorithms Laboratory St. Petersburg Institute of Informatics Russian Academy
More informationFactorization of the Robinson-Schensted-Knuth Correspondence
Factorization of the Robinson-Schensted-Knuth Correspondence David P. Little September, 00 Abstract In [], a bijection between collections of reduced factorizations of elements of the symmetric group was
More informationTableau models for Schubert polynomials
Séminaire Lotharingien de Combinatoire 78B (07) Article #, pp. Proceedings of the 9 th Conference on Formal Power Series and Algebraic Combinatorics (London) Tableau models for Schubert polynomials Sami
More informationOn Tensor Products of Polynomial Representations
Canad. Math. Bull. Vol. 5 (4), 2008 pp. 584 592 On Tensor Products of Polynomial Representations Kevin Purbhoo and Stephanie van Willigenburg Abstract. We determine the necessary and sufficient combinatorial
More informationStandard Young Tableaux Old and New
Standard Young Tableaux Old and New Ron Adin and Yuval Roichman Department of Mathematics Bar-Ilan University Workshop on Group Theory in Memory of David Chillag Technion, Haifa, Oct. 14 1 2 4 3 5 7 6
More informationarxiv: v1 [math.rt] 5 Aug 2016
AN ALGEBRAIC FORMULA FOR THE KOSTKA-FOULKES POLYNOMIALS arxiv:1608.01775v1 [math.rt] 5 Aug 2016 TIMOTHEE W. BRYAN, NAIHUAN JING Abstract. An algebraic formula for the Kostka-Foukles polynomials is given
More informationYOUNG TABLEAUX AND THE REPRESENTATIONS OF THE SYMMETRIC GROUP
YOUNG TABLEAUX AND THE REPRESENTATIONS OF THE SYMMETRIC GROUP YUFEI ZHAO ABSTRACT We explore an intimate connection between Young tableaux and representations of the symmetric group We describe the construction
More informationResults and conjectures on the number of standard strong marked tableaux
FPSAC 013, Paris, France DMTCS proc. (subm.), by the authors, 1 1 Results and conjectures on the number of standard strong marked tableaux Susanna Fishel 1 and Matjaž Konvalinka 1 School of Mathematical
More informationMultiplicity Free Expansions of Schur P-Functions
Annals of Combinatorics 11 (2007) 69-77 0218-0006/07/010069-9 DOI 10.1007/s00026-007-0306-1 c Birkhäuser Verlag, Basel, 2007 Annals of Combinatorics Multiplicity Free Expansions of Schur P-Functions Kristin
More informationA Formula for the Specialization of Skew Schur Functions
A Formula for the Specialization of Skew Schur Functions The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation As Published Publisher
More informationMULTI-ORDERED POSETS. Lisa Bishop Department of Mathematics, Occidental College, Los Angeles, CA 90041, United States.
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A06 MULTI-ORDERED POSETS Lisa Bishop Department of Mathematics, Occidental College, Los Angeles, CA 90041, United States lbishop@oxy.edu
More informationA proof of the Square Paths Conjecture
A proof of the Square Paths Conjecture Emily Sergel Leven October 7, 08 arxiv:60.069v [math.co] Jan 06 Abstract The modified Macdonald polynomials, introduced by Garsia and Haiman (996), have many astounding
More informationA note on quantum products of Schubert classes in a Grassmannian
J Algebr Comb (2007) 25:349 356 DOI 10.1007/s10801-006-0040-5 A note on quantum products of Schubert classes in a Grassmannian Dave Anderson Received: 22 August 2006 / Accepted: 14 September 2006 / Published
More informationA Pieri rule for skew shapes
A Pieri rule for skew shapes Sami H. Assaf 1 Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Peter R. W. McNamara Department of Mathematics, Bucknell University,
More informationCyclic sieving for longest reduced words in the hyperoctahedral group
FPSAC 2010, San Francisco, USA DMTCS proc. AN, 2010, 983 994 Cyclic sieving for longest reduced words in the hyperoctahedral group T. K. Petersen 1 and L. Serrano 2 1 DePaul University, Chicago, IL 2 University
More informationRow-strict quasisymmetric Schur functions
Row-strict quasisymmetric Schur functions Sarah Mason and Jeffrey Remmel Mathematics Subject Classification (010). 05E05. Keywords. quasisymmetric functions, Schur functions, omega transform. Abstract.
More informationModular representations of symmetric groups: An Overview
Modular representations of symmetric groups: An Overview Bhama Srinivasan University of Illinois at Chicago Regina, May 2012 Bhama Srinivasan (University of Illinois at Chicago) Modular Representations
More informationAnatol N. Kirillov However, the actual computation of generalized exponents has remained quite enigmatic. We repeat the words of R. Gupta []: \what ha
Proceedings of the RIMS Research Project 9 on Innite Analysis c World Scientic Publishing Company DECOMPOSITION OF SYMMETRIC AND ETERIOR POWERS OF THE ADJOINT REPRESENTATION OF gl N. UNIMODALITY OF PRINCIPAL
More informationA Relationship Between the Major Index For Tableaux and the Charge Statistic For Permutations
A Relationship Between the Major Index For Tableaux and the Charge Statistic For Permutations Kendra Killpatrick Pepperdine University Malibu, California Kendra.Killpatrick@pepperdine.edu Submitted: July
More informationTHE KNUTH RELATIONS HUAN VO
THE KNUTH RELATIONS HUAN VO Abstract We define three equivalence relations on the set of words using different means It turns out that they are the same relation Motivation Recall the algorithm of row
More informationBalanced Labellings and Schubert Polynomials. Sergey Fomin. Curtis Greene. Victor Reiner. Mark Shimozono. October 11, 1995.
Balanced Labellings and Schubert Polynomials Sergey Fomin Curtis Greene Victor Reiner Mark Shimozono October 11, 1995 Abstract We study balanced labellings of diagrams representing the inversions in a
More informationREPRESENTATION THEORY OF THE SYMMETRIC GROUP (FOLLOWING [Ful97])
REPRESENTATION THEORY OF THE SYMMETRIC GROUP (FOLLOWING [Ful97]) MICHAEL WALTER. Diagrams and Tableaux Diagrams and Tableaux. A (Young) diagram λ is a partition of a natural number n 0, which we often
More informationSOME POSITIVE DIFFERENCES OF PRODUCTS OF SCHUR FUNCTIONS
SOME POSITIVE DIFFERENCES OF PRODUCTS OF SCHUR FUNCTIONS FRANÇOIS BERGERON AND PETER MCNAMARA Abstract. The product s µ s ν of two Schur functions is one of the most famous examples of a Schur-positive
More informationThe Littlewood-Richardson Rule
REPRESENTATIONS OF THE SYMMETRIC GROUP The Littlewood-Richardson Rule Aman Barot B.Sc.(Hons.) Mathematics and Computer Science, III Year April 20, 2014 Abstract We motivate and prove the Littlewood-Richardson
More informationCatalan functions and k-schur positivity
Catalan functions and k-schur positivity Jonah Blasiak Drexel University joint work with Jennifer Morse, Anna Pun, and Dan Summers November 2018 Theorem (Haiman) Macdonald positivity conjecture The modified
More informationCombinatorial bases for representations. of the Lie superalgebra gl m n
Combinatorial bases for representations of the Lie superalgebra gl m n Alexander Molev University of Sydney Gelfand Tsetlin bases for gln Gelfand Tsetlin bases for gl n Finite-dimensional irreducible representations
More informationALGEBRAIC COMBINATORICS
ALGEBRAIC COMBINATORICS Sami Assaf & Anne Schilling A Demazure crystal construction for Schubert polynomials Volume 1, issue (018), p. 5-7. The journal
More informationJOSEPH ALFANO* Department of Mathematics, Assumption s y i P (x; y) = 0 for all r; s 0 (with r + s > 0). Computer explorations by
A BASIS FOR THE Y SUBSPACE OF DIAGONAL HARMONIC POLYNOMIALS JOSEPH ALFANO* Department of Mathematics, Assumption College 500 Salisbury Street, Worcester, Massachusetts 065-0005 ABSTRACT. The space DH n
More informationON AN INFINITE-DIMENSIONAL GROUP OVER A FINITE FIELD. A. M. Vershik, S. V. Kerov
ON AN INFINITE-DIMENSIONAL GROUP OVER A FINITE FIELD A. M. Vershik, S. V. Kerov Introduction. The asymptotic representation theory studies the behavior of representations of large classical groups and
More informationA Pieri rule for key polynomials
Séminaire Lotharingien de Combinatoire 80B (2018) Article #78, 12 pp. Proceedings of the 30 th Conference on Formal Power Series and Algebraic Combinatorics (Hanover) A Pieri rule for key polynomials Sami
More informationPIERI S FORMULA FOR GENERALIZED SCHUR POLYNOMIALS
Title Pieri's formula for generalized Schur polynomials Author(s)Numata, Yasuhide CitationJournal of Algebraic Combinatorics, 26(1): 27-45 Issue Date 2007-08 Doc RL http://hdl.handle.net/2115/33803 Rights
More informationSCHUBERT POLYNOMIALS FOR THE CLASSICAL GROUPS. Sara C. Billey
SCHUBERT POLYNOMIALS FOR THE CLASSICAL GROUPS Sara C. Billey Author supported by the National Physical Science Consortium, IBM and UCSD. 1980 Mathematics Subject Classication (1985 Revision). Primary Primary
More informationIt is well-known (cf. [2,4,5,9]) that the generating function P w() summed over all tableaux of shape = where the parts in row i are at most a i and a
Counting tableaux with row and column bounds C. Krattenthalery S. G. Mohantyz Abstract. It is well-known that the generating function for tableaux of a given skew shape with r rows where the parts in the
More informationAN ALGORITHMIC SIGN-REVERSING INVOLUTION FOR SPECIAL RIM-HOOK TABLEAUX
AN ALGORITHMIC SIGN-REVERSING INVOLUTION FOR SPECIAL RIM-HOOK TABLEAUX BRUCE E. SAGAN AND JAEJIN LEE Abstract. Eğecioğlu and Remmel [2] gave an interpretation for the entries of the inverse Kostka matrix
More informationq xk y n k. , provided k < n. (This does not hold for k n.) Give a combinatorial proof of this recurrence by means of a bijective transformation.
Math 880 Alternative Challenge Problems Fall 2016 A1. Given, n 1, show that: m1 m 2 m = ( ) n+ 1 2 1, where the sum ranges over all positive integer solutions (m 1,..., m ) of m 1 + + m = n. Give both
More informationDIFFERENTIAL POSETS SIMON RUBINSTEIN-SALZEDO
DIFFERENTIAL POSETS SIMON RUBINSTEIN-SALZEDO Abstract. In this paper, we give a sampling of the theory of differential posets, including various topics that excited me. Most of the material is taken from
More informationInvolutions by Descents/Ascents and Symmetric Integral Matrices. Alan Hoffman Fest - my hero Rutgers University September 2014
by Descents/Ascents and Symmetric Integral Matrices Richard A. Brualdi University of Wisconsin-Madison Joint work with Shi-Mei Ma: European J. Combins. (to appear) Alan Hoffman Fest - my hero Rutgers University
More informationProducts of Factorial Schur Functions
Products of Factorial Schur Functions Victor Kreiman Department of Mathematics University of Georgia Athens, GA 3060 USA vkreiman@math.uga.edu Submitted: Mar 9, 008; Accepted: Jun 10, 008; Published: Jun
More informationFactorial Schur functions via the six vertex model
Factorial Schur functions via the six vertex model Peter J. McNamara Department of Mathematics Massachusetts Institute of Technology, MA 02139, USA petermc@math.mit.edu October 31, 2009 Abstract For a
More informationBoolean Algebra and Propositional Logic
Boolean Algebra and Propositional Logic Takahiro Kato June 23, 2015 This article provides yet another characterization of Boolean algebras and, using this characterization, establishes a more direct connection
More informationBoolean Algebra and Propositional Logic
Boolean Algebra and Propositional Logic Takahiro Kato September 10, 2015 ABSTRACT. This article provides yet another characterization of Boolean algebras and, using this characterization, establishes a
More informationTHREE LECTURES ON QUASIDETERMINANTS
THREE LECTURES ON QUASIDETERMINANTS Robert Lee Wilson Department of Mathematics Rutgers University The determinant of a matrix with entries in a commutative ring is a main organizing tool in commutative
More informationA PROOF OF PIERI S FORMULA USING GENERALIZED SCHENSTED INSERTION ALGORITHM FOR RC-GRAPHS.
A PROOF OF PIERI S FORMULA USING GENERALIZED SCHENSTED INSERTION ALGORITHM FOR RC-GRAPHS. MIKHAIL KOGAN AND ABHINAV KUMAR Abstract. We provide a generalization of the Schensted insertion algorithm for
More informationKirillov-Reshetikhin Crystals and Cacti
Kirillov-Reshetikhin Crystals and Cacti Balázs Elek Cornell University, Department of Mathematics 8/1/2018 Introduction Representations Let g = sl n (C) be the Lie algebra of trace 0 matrices and V = C
More informationThe symmetric group action on rank-selected posets of injective words
The symmetric group action on rank-selected posets of injective words Christos A. Athanasiadis Department of Mathematics University of Athens Athens 15784, Hellas (Greece) caath@math.uoa.gr October 28,
More informationarxiv: v1 [math.co] 27 Aug 2014
CYCLIC SIEVING AND PLETHYSM COEFFICIENTS arxiv:1408.6484v1 [math.co] 27 Aug 2014 DAVID B RUSH Abstract. A combinatorial expression for the coefficient of the Schur function s λ in the expansion of the
More informationAdjoint Representations of the Symmetric Group
Adjoint Representations of the Symmetric Group Mahir Bilen Can 1 and Miles Jones 2 1 mahirbilencan@gmail.com 2 mej016@ucsd.edu Abstract We study the restriction to the symmetric group, S n of the adjoint
More informationMOMENTS OF INERTIA ASSOCIATED WITH THE LOZENGE TILINGS OF A HEXAGON
Séminaire Lotharingien de Combinatoire 45 001, Article B45f MOMENTS OF INERTIA ASSOCIATED WITH THE LOZENGE TILINGS OF A HEXAGON ILSE FISCHER Abstract. Consider the probability that an arbitrary chosen
More informationarxiv: v2 [math.co] 5 Apr 2017
CYCLIC SIEVING AND PLETHYSM COEFFICIENTS arxiv:1408.6484v2 [math.co] 5 Apr 2017 DAVID B RUSH Abstract. A combinatorial expression for the coefficient of the Schur function s λ in the expansion of the plethysm
More informationEnumerating rc-invariant Permutations with No Long Decreasing Subsequences
Enumerating rc-invariant Permutations with No Long Decreasing Subsequences Eric S. Egge Department of Mathematics Carleton College Northfield, MN 55057 USA eegge@carleton.edu Abstract We use the Robinson-Schensted-Knuth
More informationA Plethysm Formula for p µ (x) h λ (x) William F. Doran IV
A Plethysm Formula for p µ (x) h λ (x) William F. Doran IV Department of Mathematics California Institute of Technology Pasadena, CA 925 doran@cco.caltech.edu Submitted: September 0, 996; Accepted: May
More informationOperators on k-tableaux and the k-littlewood Richardson rule for a special case. Sarah Elizabeth Iveson
Operators on k-tableaux and the k-littlewood Richardson rule for a special case by Sarah Elizabeth Iveson A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of
More informationCopyrighted Material. Type A Weyl Group Multiple Dirichlet Series
Chapter One Type A Weyl Group Multiple Dirichlet Series We begin by defining the basic shape of the class of Weyl group multiple Dirichlet series. To do so, we choose the following parameters. Φ, a reduced
More informationA combinatorial approach to the q, t-symmetry relation in Macdonald polynomials
A combinatorial approach to the q, t-symmetry relation in Macdonald polynomials Maria Monks Gillespie Department of Mathematics University of California, Berkeley Berkeley, CA, U.S.A. monks@math.berkeley.edu
More informationLecture Notes Introduction to Cluster Algebra
Lecture Notes Introduction to Cluster Algebra Ivan C.H. Ip Update: May 16, 2017 5 Review of Root Systems In this section, let us have a brief introduction to root system and finite Lie type classification
More informationCyclic Sieving and Plethysm Coefficients
FPSAC 2015, Daejeon, South Korea DMTCS proc. FPSAC 15, 2015, 333 344 Cyclic Sieving and Plethysm Coefficients David B Rush Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA,
More informationNotes on Creation Operators
Notes on Creation Operators Mike Zabrocki August 4, 2005 This set of notes will cover sort of a how to about creation operators. Because there were several references in later talks, the appendix will
More informationREPRESENTATIONS OF S n AND GL(n, C)
REPRESENTATIONS OF S n AND GL(n, C) SEAN MCAFEE 1 outline For a given finite group G, we have that the number of irreducible representations of G is equal to the number of conjugacy classes of G Although
More informationON THE COMBINATORICS OF CRYSTAL GRAPHS, II. THE CRYSTAL COMMUTOR. 1. Introduction
ON THE COMBINATORICS OF CRYSTAL GRAPHS, II. THE CRYSTAL COMMUTOR CRISTIAN LENART Abstract. We present an explicit combinatorial realization of the commutor in the category of crystals which was first studied
More informationReduced words and a formula of Macdonald
Reduced words and a formula of Macdonald Sara Billey University of Washington Based on joint work with: Alexander Holroyd and Benjamin Young preprint arxiv:70.096 Graduate Student Combinatorics Conference
More information#A69 INTEGERS 13 (2013) OPTIMAL PRIMITIVE SETS WITH RESTRICTED PRIMES
#A69 INTEGERS 3 (203) OPTIMAL PRIMITIVE SETS WITH RESTRICTED PRIMES William D. Banks Department of Mathematics, University of Missouri, Columbia, Missouri bankswd@missouri.edu Greg Martin Department of
More informationarxiv: v2 [math.rt] 28 Aug 2018
arxiv:1808.08679v2 [math.rt] 28 Aug 2018 Tableau Correspondences and Representation Theory DIGJOY PAUL AND AMRITANSHU PRASAD The Institute of Mathematical Sciences (HBNI) Chennai ARGHYA SADHUKHAN University
More informationPieri s Formula for Generalized Schur Polynomials
Formal Power Series and Algebraic Combinatorics Séries Formelles et Combinatoire Algébrique San Diego, California 2006 Pieri s Formula for Generalized Schur Polynomials Abstract. We define a generalization
More informationCharacters and triangle generation of the simple Mathieu group M 11
SEMESTER PROJECT Characters and triangle generation of the simple Mathieu group M 11 Under the supervision of Prof. Donna Testerman Dr. Claude Marion Student: Mikaël Cavallin September 11, 2010 Contents
More informationarxiv: v1 [math.co] 2 Dec 2008
An algorithmic Littlewood-Richardson rule arxiv:08.0435v [math.co] Dec 008 Ricky Ini Liu Massachusetts Institute of Technology Cambridge, Massachusetts riliu@math.mit.edu June, 03 Abstract We introduce
More informationGENERALIZATION OF THE SIGN REVERSING INVOLUTION ON THE SPECIAL RIM HOOK TABLEAUX. Jaejin Lee
Korean J. Math. 8 (00), No., pp. 89 98 GENERALIZATION OF THE SIGN REVERSING INVOLUTION ON THE SPECIAL RIM HOOK TABLEAUX Jaejin Lee Abstract. Eğecioğlu and Remmel [] gave a combinatorial interpretation
More informationLongest element of a finite Coxeter group
Longest element of a finite Coxeter group September 10, 2015 Here we draw together some well-known properties of the (unique) longest element w in a finite Coxeter group W, with reference to theorems and
More informationDUAL IMMACULATE QUASISYMMETRIC FUNCTIONS EXPAND POSITIVELY INTO YOUNG QUASISYMMETRIC SCHUR FUNCTIONS
DUAL IMMACULATE QUASISYMMETRIC FUNCTIONS EXPAND POSITIVELY INTO YOUNG QUASISYMMETRIC SCHUR FUNCTIONS EDWARD E. ALLEN, JOSHUA HALLAM, AND SARAH K. MASON Abstract. We describe a combinatorial formula for
More informationAn Involution for the Gauss Identity
An Involution for the Gauss Identity William Y. C. Chen Center for Combinatorics Nankai University, Tianjin 300071, P. R. China Email: chenstation@yahoo.com Qing-Hu Hou Center for Combinatorics Nankai
More informationDESCENT SETS FOR SYMPLECTIC GROUPS
DESCENT SETS FOR SYMPLECTIC GROUPS MARTIN RUBEY, BRUCE E. SAGAN, AND BRUCE W. WESTBURY Abstract. The descent set of an oscillating (or up-down) tableau is introduced. This descent set plays the same role
More informationRepresentation theory
Representation theory Dr. Stuart Martin 2. Chapter 2: The Okounkov-Vershik approach These guys are Andrei Okounkov and Anatoly Vershik. The two papers appeared in 96 and 05. Here are the main steps: branching
More information5-VERTEX MODELS, GELFAND-TSETLIN PATTERNS AND SEMISTANDARD YOUNG TABLEAUX
5-VERTEX MODELS, GELFAND-TSETLIN PATTERNS AND SEMISTANDARD YOUNG TABLEAUX TANTELY A. RAKOTOARISOA 1. Introduction In statistical mechanics, one studies models based on the interconnections between thermodynamic
More informationA PRIMER ON SESQUILINEAR FORMS
A PRIMER ON SESQUILINEAR FORMS BRIAN OSSERMAN This is an alternative presentation of most of the material from 8., 8.2, 8.3, 8.4, 8.5 and 8.8 of Artin s book. Any terminology (such as sesquilinear form
More informationFrom longest increasing subsequences to Whittaker functions and random polymers
From longest increasing subsequences to Whittaker functions and random polymers Neil O Connell University of Warwick British Mathematical Colloquium, April 2, 2015 Collaborators: I. Corwin, T. Seppäläinen,
More informationA Murnaghan-Nakayama Rule for k-schur Functions
A Murnaghan-Nakayama Rule for k-schur Functions Anne Schilling (joint work with Jason Bandlow, Mike Zabrocki) University of California, Davis October 31, 2012 Outline History The Murnaghan-Nakayama rule
More informationREPRESENTATION THEORY WEEK 5. B : V V k
REPRESENTATION THEORY WEEK 5 1. Invariant forms Recall that a bilinear form on a vector space V is a map satisfying B : V V k B (cv, dw) = cdb (v, w), B (v 1 + v, w) = B (v 1, w)+b (v, w), B (v, w 1 +
More informationAffine charge and the k-bounded Pieri rule
Discrete Mathematics and Theoretical Computer Science DMTCS vol. (subm.), by the authors, Affine charge and the k-bounded Pieri rule Jennifer Morse and Anne Schilling Department of Mathematics, Drexel
More informationCombinatorics for algebraic geometers
Combinatorics for algebraic geometers Calculations in enumerative geometry Maria Monks March 17, 214 Motivation Enumerative geometry In the late 18 s, Hermann Schubert investigated problems in what is
More informationOn Multiplicity-free Products of Schur P -functions. 1 Introduction
On Multiplicity-free Products of Schur P -functions Christine Bessenrodt Fakultät für Mathematik und Physik, Leibniz Universität Hannover, Welfengarten 3067 Hannover, Germany; bessen@math.uni-hannover.de
More informationStatistical Mechanics & Enumerative Geometry:
Statistical Mechanics & Enumerative Geometry: Christian Korff (ckorff@mathsglaacuk) University Research Fellow of the Royal Society Department of Mathematics, University of Glasgow joint work with C Stroppel
More informationdescribe next. (In fact Chevalley [C] has shown a more general result probably earlier, but his paper was unpublished until recently.) For a given 2 S
ON THE MULTIPLICATION OF SCHUBERT POLYNOMIALS RUDOLF WINKEL Abstract: Finding a combinatorial rule for the multiplication of Schubert polynomials is a long standing problem. In this paper we give a combinatorial
More informationINEQUALITIES OF SYMMETRIC FUNCTIONS. 1. Introduction to Symmetric Functions [?] Definition 1.1. A symmetric function in n variables is a function, f,
INEQUALITIES OF SMMETRIC FUNCTIONS JONATHAN D. LIMA Abstract. We prove several symmetric function inequalities and conjecture a partially proved comprehensive theorem. We also introduce the condition of
More informationA multiplicative deformation of the Möbius function for the poset of partitions of a multiset
Contemporary Mathematics A multiplicative deformation of the Möbius function for the poset of partitions of a multiset Patricia Hersh and Robert Kleinberg Abstract. The Möbius function of a partially ordered
More informationQUASISYMMETRIC (k, l)-hook SCHUR FUNCTIONS
QUASISYMMETRIC (k, l)-hook SCHUR FUNCTIONS SARAH K. MASON AND ELIZABETH NIESE Abstract. We introduce a quasisymmetric generalization of Berele and Regev s hook Schur functions and prove that these new
More information